| L(s) = 1 | + 6·3-s + 5·5-s − 7·7-s − 52·9-s + 13·11-s − 72·13-s + 30·15-s − 59·17-s − 95·19-s − 42·21-s − 52·23-s − 343·25-s − 420·27-s − 128·29-s + 110·31-s + 78·33-s − 35·35-s − 436·37-s − 432·39-s − 804·41-s − 143·43-s − 260·45-s − 661·47-s − 1.03e3·49-s − 354·51-s − 898·53-s + 65·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s − 1.92·9-s + 0.356·11-s − 1.53·13-s + 0.516·15-s − 0.841·17-s − 1.14·19-s − 0.436·21-s − 0.471·23-s − 2.74·25-s − 2.99·27-s − 0.819·29-s + 0.637·31-s + 0.411·33-s − 0.169·35-s − 1.93·37-s − 1.77·39-s − 3.06·41-s − 0.507·43-s − 0.861·45-s − 2.05·47-s − 3.02·49-s − 0.971·51-s − 2.32·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 - 2 p T + 88 T^{2} - 140 p T^{3} + 3979 T^{4} - 5252 p T^{5} + 3979 p^{3} T^{6} - 140 p^{7} T^{7} + 88 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - p T + 368 T^{2} - 27 p^{2} T^{3} + 70303 T^{4} - 18532 p T^{5} + 70303 p^{3} T^{6} - 27 p^{8} T^{7} + 368 p^{9} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + p T + 155 p T^{2} + 12674 T^{3} + 544465 T^{4} + 7054697 T^{5} + 544465 p^{3} T^{6} + 12674 p^{6} T^{7} + 155 p^{10} T^{8} + p^{13} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 13 T + 4710 T^{2} - 22007 T^{3} + 9702469 T^{4} - 12644628 T^{5} + 9702469 p^{3} T^{6} - 22007 p^{6} T^{7} + 4710 p^{9} T^{8} - 13 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 72 T + 7692 T^{2} + 452708 T^{3} + 27305959 T^{4} + 1294665496 T^{5} + 27305959 p^{3} T^{6} + 452708 p^{6} T^{7} + 7692 p^{9} T^{8} + 72 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 59 T + 13187 T^{2} + 999910 T^{3} + 101512197 T^{4} + 6679234593 T^{5} + 101512197 p^{3} T^{6} + 999910 p^{6} T^{7} + 13187 p^{9} T^{8} + 59 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 52 T + 50854 T^{2} + 2342144 T^{3} + 1131064489 T^{4} + 41902894872 T^{5} + 1131064489 p^{3} T^{6} + 2342144 p^{6} T^{7} + 50854 p^{9} T^{8} + 52 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 128 T + 36692 T^{2} + 3212 p T^{3} + 529605375 T^{4} - 40988606664 T^{5} + 529605375 p^{3} T^{6} + 3212 p^{7} T^{7} + 36692 p^{9} T^{8} + 128 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 110 T + 81071 T^{2} - 7894736 T^{3} + 3729288174 T^{4} - 328269356932 T^{5} + 3729288174 p^{3} T^{6} - 7894736 p^{6} T^{7} + 81071 p^{9} T^{8} - 110 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 436 T + 298973 T^{2} + 86764576 T^{3} + 32369632686 T^{4} + 6556844678168 T^{5} + 32369632686 p^{3} T^{6} + 86764576 p^{6} T^{7} + 298973 p^{9} T^{8} + 436 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 804 T + 369149 T^{2} + 110044272 T^{3} + 27603772810 T^{4} + 6675353551896 T^{5} + 27603772810 p^{3} T^{6} + 110044272 p^{6} T^{7} + 369149 p^{9} T^{8} + 804 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 143 T + 204630 T^{2} + 15645817 T^{3} + 22191909365 T^{4} + 877925846376 T^{5} + 22191909365 p^{3} T^{6} + 15645817 p^{6} T^{7} + 204630 p^{9} T^{8} + 143 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 661 T + 443686 T^{2} + 2703165 p T^{3} + 46701653905 T^{4} + 9160432775440 T^{5} + 46701653905 p^{3} T^{6} + 2703165 p^{7} T^{7} + 443686 p^{9} T^{8} + 661 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 898 T + 963008 T^{2} + 539672346 T^{3} + 316583941975 T^{4} + 120708500335000 T^{5} + 316583941975 p^{3} T^{6} + 539672346 p^{6} T^{7} + 963008 p^{9} T^{8} + 898 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 196 T + 830928 T^{2} - 135237746 T^{3} + 302800264447 T^{4} - 39146939158716 T^{5} + 302800264447 p^{3} T^{6} - 135237746 p^{6} T^{7} + 830928 p^{9} T^{8} - 196 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 1079 T + 1333048 T^{2} + 881544597 T^{3} + 615902886183 T^{4} + 286356191273776 T^{5} + 615902886183 p^{3} T^{6} + 881544597 p^{6} T^{7} + 1333048 p^{9} T^{8} + 1079 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 832 T + 816120 T^{2} - 572054486 T^{3} + 317286443447 T^{4} - 194405950061388 T^{5} + 317286443447 p^{3} T^{6} - 572054486 p^{6} T^{7} + 816120 p^{9} T^{8} - 832 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 834 T + 1701275 T^{2} + 1054732616 T^{3} + 1163286176194 T^{4} + 540650010599884 T^{5} + 1163286176194 p^{3} T^{6} + 1054732616 p^{6} T^{7} + 1701275 p^{9} T^{8} + 834 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 1375 T + 2328039 T^{2} + 2029439450 T^{3} + 1924299315773 T^{4} + 1165512957784425 T^{5} + 1924299315773 p^{3} T^{6} + 2029439450 p^{6} T^{7} + 2328039 p^{9} T^{8} + 1375 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 154 T + 1421235 T^{2} + 366265736 T^{3} + 1077036759682 T^{4} + 269105543902940 T^{5} + 1077036759682 p^{3} T^{6} + 366265736 p^{6} T^{7} + 1421235 p^{9} T^{8} + 154 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 2224 T + 3136947 T^{2} - 2898613184 T^{3} + 2188724101174 T^{4} - 1562584349928480 T^{5} + 2188724101174 p^{3} T^{6} - 2898613184 p^{6} T^{7} + 3136947 p^{9} T^{8} - 2224 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 542 T + 1004081 T^{2} + 825027904 T^{3} + 844541522422 T^{4} + 1039222693803396 T^{5} + 844541522422 p^{3} T^{6} + 825027904 p^{6} T^{7} + 1004081 p^{9} T^{8} + 542 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 1528 T + 4223497 T^{2} + 5358794000 T^{3} + 7393031526134 T^{4} + 7236303469568816 T^{5} + 7393031526134 p^{3} T^{6} + 5358794000 p^{6} T^{7} + 4223497 p^{9} T^{8} + 1528 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43864115170584110685367840201, −6.31625169843796775730669192301, −6.28169723645178176940923361890, −6.08835354685652503481126197117, −5.92878137447091350867231736083, −5.35446245632999007283173180894, −5.32455961106673141695972062395, −5.30479276709694853848187341091, −4.92313746841813392491061486715, −4.80688935017697010948008845773, −4.75429356538495847910011215379, −4.07871224843389709113405789698, −4.07509333081071978830525758415, −3.75885781153831741646007262232, −3.74659313720450847760137994827, −3.21353255774194248801428303076, −3.15722967927310516109032309352, −2.90792698371701395340766780130, −2.70607830682827142597786644160, −2.68762737406417200615796665213, −1.96032296430138311498196215004, −1.94333165002042830225112217241, −1.86545654666441810485161251732, −1.53530547015177849487995772050, −1.33533005742931618238472936527, 0, 0, 0, 0, 0,
1.33533005742931618238472936527, 1.53530547015177849487995772050, 1.86545654666441810485161251732, 1.94333165002042830225112217241, 1.96032296430138311498196215004, 2.68762737406417200615796665213, 2.70607830682827142597786644160, 2.90792698371701395340766780130, 3.15722967927310516109032309352, 3.21353255774194248801428303076, 3.74659313720450847760137994827, 3.75885781153831741646007262232, 4.07509333081071978830525758415, 4.07871224843389709113405789698, 4.75429356538495847910011215379, 4.80688935017697010948008845773, 4.92313746841813392491061486715, 5.30479276709694853848187341091, 5.32455961106673141695972062395, 5.35446245632999007283173180894, 5.92878137447091350867231736083, 6.08835354685652503481126197117, 6.28169723645178176940923361890, 6.31625169843796775730669192301, 6.43864115170584110685367840201
