| L(s) = 1 | + 10·2-s + 60·4-s + 20·7-s + 280·8-s − 62·9-s + 16·11-s + 56·13-s + 200·14-s + 1.12e3·16-s + 70·17-s − 620·18-s + 32·19-s + 160·22-s − 115·23-s + 560·26-s + 73·27-s + 1.20e3·28-s + 128·29-s + 86·31-s + 4.03e3·32-s + 700·34-s − 3.72e3·36-s + 346·37-s + 320·38-s − 102·41-s + 154·43-s + 960·44-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 15/2·4-s + 1.07·7-s + 12.3·8-s − 2.29·9-s + 0.438·11-s + 1.19·13-s + 3.81·14-s + 35/2·16-s + 0.998·17-s − 8.11·18-s + 0.386·19-s + 1.55·22-s − 1.04·23-s + 4.22·26-s + 0.520·27-s + 8.09·28-s + 0.819·29-s + 0.498·31-s + 22.2·32-s + 3.53·34-s − 17.2·36-s + 1.53·37-s + 1.36·38-s − 0.388·41-s + 0.546·43-s + 3.28·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(249.5591596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(249.5591596\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{5} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 62 T^{2} - 73 T^{3} + p^{7} T^{4} - 4202 T^{5} + p^{10} T^{6} - 73 p^{6} T^{7} + 62 p^{9} T^{8} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 20 T + 922 T^{2} - 13740 T^{3} + 69487 p T^{4} - 962600 p T^{5} + 69487 p^{4} T^{6} - 13740 p^{6} T^{7} + 922 p^{9} T^{8} - 20 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 16 T + 4212 T^{2} - 76666 T^{3} + 790397 p T^{4} - 143535476 T^{5} + 790397 p^{4} T^{6} - 76666 p^{6} T^{7} + 4212 p^{9} T^{8} - 16 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 56 T + 6421 T^{2} - 324423 T^{3} + 23069197 T^{4} - 971185271 T^{5} + 23069197 p^{3} T^{6} - 324423 p^{6} T^{7} + 6421 p^{9} T^{8} - 56 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 70 T + 17117 T^{2} - 565472 T^{3} + 108600418 T^{4} - 2000023092 T^{5} + 108600418 p^{3} T^{6} - 565472 p^{6} T^{7} + 17117 p^{9} T^{8} - 70 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 32 T + 15004 T^{2} - 871122 T^{3} + 175059199 T^{4} - 5937831716 T^{5} + 175059199 p^{3} T^{6} - 871122 p^{6} T^{7} + 15004 p^{9} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 128 T + 93901 T^{2} - 8087785 T^{3} + 3790762021 T^{4} - 241558234065 T^{5} + 3790762021 p^{3} T^{6} - 8087785 p^{6} T^{7} + 93901 p^{9} T^{8} - 128 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 86 T + 106072 T^{2} - 10826487 T^{3} + 5226516991 T^{4} - 488728701242 T^{5} + 5226516991 p^{3} T^{6} - 10826487 p^{6} T^{7} + 106072 p^{9} T^{8} - 86 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 346 T + 219641 T^{2} - 59998632 T^{3} + 21673704690 T^{4} - 4277811446236 T^{5} + 21673704690 p^{3} T^{6} - 59998632 p^{6} T^{7} + 219641 p^{9} T^{8} - 346 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 102 T + 316107 T^{2} + 27534699 T^{3} + 41678281315 T^{4} + 2825442259599 T^{5} + 41678281315 p^{3} T^{6} + 27534699 p^{6} T^{7} + 316107 p^{9} T^{8} + 102 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 154 T + 333812 T^{2} - 37015464 T^{3} + 1112603589 p T^{4} - 3957701966044 T^{5} + 1112603589 p^{4} T^{6} - 37015464 p^{6} T^{7} + 333812 p^{9} T^{8} - 154 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 906 T + 451380 T^{2} - 137943657 T^{3} + 30249265825 T^{4} - 7348186021974 T^{5} + 30249265825 p^{3} T^{6} - 137943657 p^{6} T^{7} + 451380 p^{9} T^{8} - 906 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 142 T + 634805 T^{2} + 78475928 T^{3} + 174754735558 T^{4} + 17199372456948 T^{5} + 174754735558 p^{3} T^{6} + 78475928 p^{6} T^{7} + 634805 p^{9} T^{8} + 142 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 1443 T + 1585852 T^{2} - 1136197797 T^{3} + 705281058433 T^{4} - 337617714243420 T^{5} + 705281058433 p^{3} T^{6} - 1136197797 p^{6} T^{7} + 1585852 p^{9} T^{8} - 1443 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 102 T + 804609 T^{2} + 27638376 T^{3} + 307485028178 T^{4} + 5685627582468 T^{5} + 307485028178 p^{3} T^{6} + 27638376 p^{6} T^{7} + 804609 p^{9} T^{8} + 102 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1000 T + 1378427 T^{2} - 746024472 T^{3} + 614986858158 T^{4} - 246933645956032 T^{5} + 614986858158 p^{3} T^{6} - 746024472 p^{6} T^{7} + 1378427 p^{9} T^{8} - 1000 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 384 T + 1535520 T^{2} + 381074373 T^{3} + 976405450675 T^{4} + 172491171293418 T^{5} + 976405450675 p^{3} T^{6} + 381074373 p^{6} T^{7} + 1535520 p^{9} T^{8} + 384 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 90 T + 1366007 T^{2} - 175127279 T^{3} + 925988284399 T^{4} - 92499502829707 T^{5} + 925988284399 p^{3} T^{6} - 175127279 p^{6} T^{7} + 1366007 p^{9} T^{8} - 90 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 1768 T + 3353090 T^{2} - 3507404436 T^{3} + 3687457407573 T^{4} - 2597940599143240 T^{5} + 3687457407573 p^{3} T^{6} - 3507404436 p^{6} T^{7} + 3353090 p^{9} T^{8} - 1768 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 1278 T + 2149612 T^{2} - 2144734640 T^{3} + 2171389744123 T^{4} - 1673876240227244 T^{5} + 2171389744123 p^{3} T^{6} - 2144734640 p^{6} T^{7} + 2149612 p^{9} T^{8} - 1278 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 1970 T + 4061633 T^{2} - 5110884568 T^{3} + 5988742854598 T^{4} - 5277531848765292 T^{5} + 5988742854598 p^{3} T^{6} - 5110884568 p^{6} T^{7} + 4061633 p^{9} T^{8} - 1970 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 310 T + 1542485 T^{2} + 447328920 T^{3} + 1428112636890 T^{4} + 683321295875132 T^{5} + 1428112636890 p^{3} T^{6} + 447328920 p^{6} T^{7} + 1542485 p^{9} T^{8} - 310 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.37746330209579065479532981181, −5.25392468077156245337347559867, −5.17201763603726449550137631776, −5.08700381793897210588861012264, −4.87093549157847289394366409617, −4.45243868474491700766849890406, −4.33603825406578606948782789392, −4.29880704370458322180646448999, −4.03378011220318611461521949255, −3.92676130341575681337983932542, −3.46355889080499659839392080473, −3.34191059003621828205307661873, −3.26668949147155473072078289036, −3.20650870741651640532280523974, −3.05059655809768305140073978633, −2.36400390092089069453280008812, −2.33922854966109417740171928651, −2.30194647958067869208396321089, −2.19008481890594282829575222477, −1.86512492188450688213174091682, −1.38685422443502002898226357103, −1.06084045940530884392876786314, −0.859451858066948514173649778707, −0.790187343120567290683312704769, −0.42172709134101521626140800846,
0.42172709134101521626140800846, 0.790187343120567290683312704769, 0.859451858066948514173649778707, 1.06084045940530884392876786314, 1.38685422443502002898226357103, 1.86512492188450688213174091682, 2.19008481890594282829575222477, 2.30194647958067869208396321089, 2.33922854966109417740171928651, 2.36400390092089069453280008812, 3.05059655809768305140073978633, 3.20650870741651640532280523974, 3.26668949147155473072078289036, 3.34191059003621828205307661873, 3.46355889080499659839392080473, 3.92676130341575681337983932542, 4.03378011220318611461521949255, 4.29880704370458322180646448999, 4.33603825406578606948782789392, 4.45243868474491700766849890406, 4.87093549157847289394366409617, 5.08700381793897210588861012264, 5.17201763603726449550137631776, 5.25392468077156245337347559867, 5.37746330209579065479532981181
