| L(s) = 1 | − 8·3-s − 7·4-s − 10·5-s − 28·8-s − 19·9-s + 12·11-s + 56·12-s − 14·13-s + 80·15-s + 9·16-s − 66·17-s − 214·19-s + 70·20-s + 164·23-s + 224·24-s − 159·25-s + 274·27-s − 145·29-s − 420·31-s + 200·32-s − 96·33-s + 133·36-s + 378·37-s + 112·39-s + 280·40-s + 1.15e3·41-s − 204·43-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 7/8·4-s − 0.894·5-s − 1.23·8-s − 0.703·9-s + 0.328·11-s + 1.34·12-s − 0.298·13-s + 1.37·15-s + 9/64·16-s − 0.941·17-s − 2.58·19-s + 0.782·20-s + 1.48·23-s + 1.90·24-s − 1.27·25-s + 1.95·27-s − 0.928·29-s − 2.43·31-s + 1.10·32-s − 0.506·33-s + 0.615·36-s + 1.67·37-s + 0.459·39-s + 1.10·40-s + 4.41·41-s − 0.723·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + 7 T^{2} + 7 p^{2} T^{3} + 5 p^{3} T^{4} + 3 p^{6} T^{5} + 5 p^{6} T^{6} + 7 p^{8} T^{7} + 7 p^{9} T^{8} + p^{15} T^{10} \) |
| 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 | $C_2 \wr S_5$ | \( 1 + 2 p T + 259 T^{2} + 2096 T^{3} + 40453 T^{4} + 267034 T^{5} + 40453 p^{3} T^{6} + 2096 p^{6} T^{7} + 259 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} \) |
| 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 | $C_2 \wr S_5$ | \( 1 + 1562 T + 4976353 T^{2} + 5411957216 T^{3} + 9379455939582 T^{4} + 7296970015174028 T^{5} + 9379455939582 p^{3} T^{6} + 5411957216 p^{6} T^{7} + 4976353 p^{9} T^{8} + 1562 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.88077681926991045064208733285, −5.64394618447451274361239646533, −5.36869201482141272571211050294, −5.32306825531168414029152127618, −5.29469098033087521474705774759, −4.76438968089673336202124356172, −4.70398813080088189051034884935, −4.69844516479047676166513800649, −4.27427196030925584701096633708, −4.13371177924014846289952134281, −4.10199896253598119574887031856, −3.79613537310271263041257020249, −3.61670294164068066228932334707, −3.53269092498805101029208216751, −3.11752676325919430580275549327, −3.10283506230013432610621816117, −2.67426477312413567365035490289, −2.37115191886809887755433751136, −2.29078491845076573109712208150, −2.17036198579968363593520840930, −2.10326644487304815857558401617, −1.36975370291910841160564044045, −1.13095939722162695104817925737, −1.06905004813252945644876899645, −0.63829289487689418652528604047, 0, 0, 0, 0, 0,
0.63829289487689418652528604047, 1.06905004813252945644876899645, 1.13095939722162695104817925737, 1.36975370291910841160564044045, 2.10326644487304815857558401617, 2.17036198579968363593520840930, 2.29078491845076573109712208150, 2.37115191886809887755433751136, 2.67426477312413567365035490289, 3.10283506230013432610621816117, 3.11752676325919430580275549327, 3.53269092498805101029208216751, 3.61670294164068066228932334707, 3.79613537310271263041257020249, 4.10199896253598119574887031856, 4.13371177924014846289952134281, 4.27427196030925584701096633708, 4.69844516479047676166513800649, 4.70398813080088189051034884935, 4.76438968089673336202124356172, 5.29469098033087521474705774759, 5.32306825531168414029152127618, 5.36869201482141272571211050294, 5.64394618447451274361239646533, 5.88077681926991045064208733285
