| L(s) = 1 | + 2·5-s − 7·7-s + 11-s + 5·13-s + 3·17-s − 3·19-s + 3·23-s − 5·25-s + 12·29-s − 8·31-s − 14·35-s + 5·37-s + 10·41-s − 8·43-s + 24·47-s + 14·49-s + 13·53-s + 2·55-s + 18·59-s + 10·61-s + 10·65-s − 4·67-s + 24·71-s + 13·73-s − 7·77-s − 22·79-s + 25·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.64·7-s + 0.301·11-s + 1.38·13-s + 0.727·17-s − 0.688·19-s + 0.625·23-s − 25-s + 2.22·29-s − 1.43·31-s − 2.36·35-s + 0.821·37-s + 1.56·41-s − 1.21·43-s + 3.50·47-s + 2·49-s + 1.78·53-s + 0.269·55-s + 2.34·59-s + 1.28·61-s + 1.24·65-s − 0.488·67-s + 2.84·71-s + 1.52·73-s − 0.797·77-s − 2.47·79-s + 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.618679275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.618679275\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 5 | $C_2 \wr S_5$ | \( 1 - 2 T + 9 T^{2} - 4 p T^{3} + 78 T^{4} - 124 T^{5} + 78 p T^{6} - 4 p^{3} T^{7} + 9 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + p T + 5 p T^{2} + 132 T^{3} + 458 T^{4} + 1306 T^{5} + 458 p T^{6} + 132 p^{2} T^{7} + 5 p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - T + 19 T^{2} - 4 p T^{3} + 342 T^{4} - 342 T^{5} + 342 p T^{6} - 4 p^{3} T^{7} + 19 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 5 T + 33 T^{2} - 12 p T^{3} + 618 T^{4} - 2318 T^{5} + 618 p T^{6} - 12 p^{3} T^{7} + 33 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 3 T + 33 T^{2} - 8 T^{3} + 526 T^{4} + 210 T^{5} + 526 p T^{6} - 8 p^{2} T^{7} + 33 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 3 T + 59 T^{2} + 212 T^{3} + 1766 T^{4} + 5778 T^{5} + 1766 p T^{6} + 212 p^{2} T^{7} + 59 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 3 T + 91 T^{2} - 200 T^{3} + 3714 T^{4} - 6278 T^{5} + 3714 p T^{6} - 200 p^{2} T^{7} + 91 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 12 T + 5 p T^{2} - 1116 T^{3} + 8230 T^{4} - 44576 T^{5} + 8230 p T^{6} - 1116 p^{2} T^{7} + 5 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 8 T + 123 T^{2} + 800 T^{3} + 6426 T^{4} + 34160 T^{5} + 6426 p T^{6} + 800 p^{2} T^{7} + 123 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 10 T + 181 T^{2} - 1192 T^{3} + 12898 T^{4} - 64252 T^{5} + 12898 p T^{6} - 1192 p^{2} T^{7} + 181 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 8 T + 7 T^{2} - 96 T^{3} + 2010 T^{4} + 20976 T^{5} + 2010 p T^{6} - 96 p^{2} T^{7} + 7 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 24 T + 411 T^{2} - 4832 T^{3} + 46458 T^{4} - 347920 T^{5} + 46458 p T^{6} - 4832 p^{2} T^{7} + 411 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 13 T + 197 T^{2} - 1668 T^{3} + 18302 T^{4} - 126430 T^{5} + 18302 p T^{6} - 1668 p^{2} T^{7} + 197 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 18 T + 319 T^{2} - 3836 T^{3} + 37486 T^{4} - 325756 T^{5} + 37486 p T^{6} - 3836 p^{2} T^{7} + 319 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{5} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 4 T + 175 T^{2} + 768 T^{3} + 15978 T^{4} + 66808 T^{5} + 15978 p T^{6} + 768 p^{2} T^{7} + 175 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 24 T + 531 T^{2} - 7136 T^{3} + 87450 T^{4} - 771088 T^{5} + 87450 p T^{6} - 7136 p^{2} T^{7} + 531 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 13 T + 325 T^{2} - 3140 T^{3} + 44994 T^{4} - 320110 T^{5} + 44994 p T^{6} - 3140 p^{2} T^{7} + 325 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 22 T + 483 T^{2} + 6696 T^{3} + 83442 T^{4} + 783332 T^{5} + 83442 p T^{6} + 6696 p^{2} T^{7} + 483 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 25 T + 455 T^{2} - 5964 T^{3} + 67554 T^{4} - 648966 T^{5} + 67554 p T^{6} - 5964 p^{2} T^{7} + 455 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 27 T + 629 T^{2} - 9072 T^{3} + 119514 T^{4} - 13102 p T^{5} + 119514 p T^{6} - 9072 p^{2} T^{7} + 629 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 20 T + 477 T^{2} - 6288 T^{3} + 88578 T^{4} - 859064 T^{5} + 88578 p T^{6} - 6288 p^{2} T^{7} + 477 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) |
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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
−5.31831124838661156647411654842, −5.11620797666622956580874026945, −4.89633653025997335320013620917, −4.87439520679944419932339010922, −4.78158404834047202204560301246, −4.15197149228899984149627423272, −4.05019408166106842767043933473, −3.94129555546909072275074497655, −3.92086265269222323941865138427, −3.90145258189997707863476773870, −3.41008948629845512181075181318, −3.30620289141839629288988844997, −3.22610152493788464115476599046, −3.00597659177097025759381076414, −2.87275015259803319761765672185, −2.35130605811605934234486712290, −2.26661215137658693505206107340, −2.20082193095866099539715802491, −2.13168239314445263726884620584, −1.85363536587570713630774880696, −1.12533219577381985441304266977, −1.10588166152722100365828003701, −0.886206285444425301741639805532, −0.57908031945515831172735156982, −0.45859705460120520279036375400,
0.45859705460120520279036375400, 0.57908031945515831172735156982, 0.886206285444425301741639805532, 1.10588166152722100365828003701, 1.12533219577381985441304266977, 1.85363536587570713630774880696, 2.13168239314445263726884620584, 2.20082193095866099539715802491, 2.26661215137658693505206107340, 2.35130605811605934234486712290, 2.87275015259803319761765672185, 3.00597659177097025759381076414, 3.22610152493788464115476599046, 3.30620289141839629288988844997, 3.41008948629845512181075181318, 3.90145258189997707863476773870, 3.92086265269222323941865138427, 3.94129555546909072275074497655, 4.05019408166106842767043933473, 4.15197149228899984149627423272, 4.78158404834047202204560301246, 4.87439520679944419932339010922, 4.89633653025997335320013620917, 5.11620797666622956580874026945, 5.31831124838661156647411654842
