| L(s) = 1 | − 8·3-s − 10·5-s + 40·7-s − 19·9-s − 12·11-s − 14·13-s + 80·15-s + 66·17-s − 214·19-s − 320·21-s + 164·23-s − 159·25-s + 274·27-s + 145·29-s + 420·31-s + 96·33-s − 400·35-s − 378·37-s + 112·39-s − 1.15e3·41-s + 204·43-s + 190·45-s + 248·47-s − 199·49-s − 528·51-s + 554·53-s + 120·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.894·5-s + 2.15·7-s − 0.703·9-s − 0.328·11-s − 0.298·13-s + 1.37·15-s + 0.941·17-s − 2.58·19-s − 3.32·21-s + 1.48·23-s − 1.27·25-s + 1.95·27-s + 0.928·29-s + 2.43·31-s + 0.506·33-s − 1.93·35-s − 1.67·37-s + 0.459·39-s − 4.41·41-s + 0.723·43-s + 0.629·45-s + 0.769·47-s − 0.580·49-s − 1.44·51-s + 1.43·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 8 T + 83 T^{2} + 542 T^{3} + 3265 T^{4} + 18646 T^{5} + 3265 p^{3} T^{6} + 542 p^{6} T^{7} + 83 p^{9} T^{8} + 8 p^{12} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 2 p T + 259 T^{2} + 2096 T^{3} + 40453 T^{4} + 267034 T^{5} + 40453 p^{3} T^{6} + 2096 p^{6} T^{7} + 259 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 40 T + 257 p T^{2} - 44112 T^{3} + 1162638 T^{4} - 20604944 T^{5} + 1162638 p^{3} T^{6} - 44112 p^{6} T^{7} + 257 p^{10} T^{8} - 40 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 12 T + 1763 T^{2} + 13714 T^{3} + 2580641 T^{4} - 37008754 T^{5} + 2580641 p^{3} T^{6} + 13714 p^{6} T^{7} + 1763 p^{9} T^{8} + 12 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 14 T + 3427 T^{2} - 10280 T^{3} - 252123 T^{4} - 167242554 T^{5} - 252123 p^{3} T^{6} - 10280 p^{6} T^{7} + 3427 p^{9} T^{8} + 14 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 66 T + 22121 T^{2} - 1091584 T^{3} + 201659462 T^{4} - 7519809404 T^{5} + 201659462 p^{3} T^{6} - 1091584 p^{6} T^{7} + 22121 p^{9} T^{8} - 66 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 214 T + 49431 T^{2} + 6213576 T^{3} + 780820954 T^{4} + 65082757348 T^{5} + 780820954 p^{3} T^{6} + 6213576 p^{6} T^{7} + 49431 p^{9} T^{8} + 214 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 164 T + 42023 T^{2} - 4665552 T^{3} + 817883710 T^{4} - 72913879960 T^{5} + 817883710 p^{3} T^{6} - 4665552 p^{6} T^{7} + 42023 p^{9} T^{8} - 164 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 420 T + 194507 T^{2} - 51412234 T^{3} + 12940581345 T^{4} - 2317742728494 T^{5} + 12940581345 p^{3} T^{6} - 51412234 p^{6} T^{7} + 194507 p^{9} T^{8} - 420 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 378 T + 183473 T^{2} + 45668424 T^{3} + 12598747514 T^{4} + 2663232080572 T^{5} + 12598747514 p^{3} T^{6} + 45668424 p^{6} T^{7} + 183473 p^{9} T^{8} + 378 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 1158 T + 807513 T^{2} + 393665952 T^{3} + 147623041214 T^{4} + 43322074461428 T^{5} + 147623041214 p^{3} T^{6} + 393665952 p^{6} T^{7} + 807513 p^{9} T^{8} + 1158 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 204 T + 303147 T^{2} - 53162326 T^{3} + 42555587009 T^{4} - 6073081393458 T^{5} + 42555587009 p^{3} T^{6} - 53162326 p^{6} T^{7} + 303147 p^{9} T^{8} - 204 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 248 T + 186419 T^{2} - 26483122 T^{3} + 12346963729 T^{4} - 356058792474 T^{5} + 12346963729 p^{3} T^{6} - 26483122 p^{6} T^{7} + 186419 p^{9} T^{8} - 248 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 554 T + 706051 T^{2} - 257171096 T^{3} + 190377537541 T^{4} - 51228756286434 T^{5} + 190377537541 p^{3} T^{6} - 257171096 p^{6} T^{7} + 706051 p^{9} T^{8} - 554 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 440 T + 931027 T^{2} + 318478640 T^{3} + 358630287294 T^{4} + 93589746901040 T^{5} + 358630287294 p^{3} T^{6} + 318478640 p^{6} T^{7} + 931027 p^{9} T^{8} + 440 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 618 T + 930749 T^{2} + 354654304 T^{3} + 335118575446 T^{4} + 95179696680076 T^{5} + 335118575446 p^{3} T^{6} + 354654304 p^{6} T^{7} + 930749 p^{9} T^{8} + 618 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 1164 T + 1251831 T^{2} + 943021136 T^{3} + 702489951274 T^{4} + 395972688310152 T^{5} + 702489951274 p^{3} T^{6} + 943021136 p^{6} T^{7} + 1251831 p^{9} T^{8} + 1164 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 692 T + 912707 T^{2} + 418495168 T^{3} + 492751147498 T^{4} + 220618574155288 T^{5} + 492751147498 p^{3} T^{6} + 418495168 p^{6} T^{7} + 912707 p^{9} T^{8} + 692 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 1950 T + 3113117 T^{2} + 3302203032 T^{3} + 2895801534794 T^{4} + 1979029953150004 T^{5} + 2895801534794 p^{3} T^{6} + 3302203032 p^{6} T^{7} + 3113117 p^{9} T^{8} + 1950 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 272 T + 967307 T^{2} + 8685042 T^{3} + 758497584129 T^{4} - 100158281651838 T^{5} + 758497584129 p^{3} T^{6} + 8685042 p^{6} T^{7} + 967307 p^{9} T^{8} - 272 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 512 T + 2338779 T^{2} + 1052487024 T^{3} + 2462969890974 T^{4} + 862752231073696 T^{5} + 2462969890974 p^{3} T^{6} + 1052487024 p^{6} T^{7} + 2338779 p^{9} T^{8} + 512 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 866 T + 2672425 T^{2} - 2372819264 T^{3} + 3305080150846 T^{4} - 2463206457135612 T^{5} + 3305080150846 p^{3} T^{6} - 2372819264 p^{6} T^{7} + 2672425 p^{9} T^{8} - 866 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 1562 T + 4976353 T^{2} - 5411957216 T^{3} + 9379455939582 T^{4} - 7296970015174028 T^{5} + 9379455939582 p^{3} T^{6} - 5411957216 p^{6} T^{7} + 4976353 p^{9} T^{8} - 1562 p^{12} T^{9} + p^{15} T^{10} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.32049357261464651929409068216, −5.27714686717303473364204371403, −5.15489385806100733276816374516, −5.08996763030716258396747875615, −5.08622840736103304014441220342, −4.86496192334696858442598665179, −4.53369169869112976909313680943, −4.38095073200934793466793782573, −4.31446204667673888821781915795, −4.24090902312402787142243518065, −3.81267031600309620110055640095, −3.60369921260683542776789615104, −3.37650637131269261840149172013, −3.24658036428366790454104490919, −3.13394496298783334510622350428, −2.78813516098730040325027745801, −2.52629396243925500829846400147, −2.44448436074356498566047688829, −2.22384003129698242774997197889, −1.86844166743318844357118936195, −1.62754872869830660272071734016, −1.41600545151098381940867217950, −1.31372328702898687549741934268, −1.07503721291844775694128717517, −0.820125476183439749958637940278, 0, 0, 0, 0, 0,
0.820125476183439749958637940278, 1.07503721291844775694128717517, 1.31372328702898687549741934268, 1.41600545151098381940867217950, 1.62754872869830660272071734016, 1.86844166743318844357118936195, 2.22384003129698242774997197889, 2.44448436074356498566047688829, 2.52629396243925500829846400147, 2.78813516098730040325027745801, 3.13394496298783334510622350428, 3.24658036428366790454104490919, 3.37650637131269261840149172013, 3.60369921260683542776789615104, 3.81267031600309620110055640095, 4.24090902312402787142243518065, 4.31446204667673888821781915795, 4.38095073200934793466793782573, 4.53369169869112976909313680943, 4.86496192334696858442598665179, 5.08622840736103304014441220342, 5.08996763030716258396747875615, 5.15489385806100733276816374516, 5.27714686717303473364204371403, 5.32049357261464651929409068216
