| L(s) = 1 | + 105·5-s + 937·7-s − 5.94e3·11-s + 68·13-s − 5.40e3·17-s + 4.83e4·19-s + 642·23-s − 6.71e4·25-s − 1.25e5·29-s + 1.61e5·31-s + 9.83e4·35-s − 4.14e5·37-s − 6.27e5·41-s − 5.70e5·43-s − 5.38e5·47-s + 5.44e4·49-s + 3.56e5·53-s − 6.24e5·55-s + 2.91e6·59-s + 2.68e6·61-s + 7.14e3·65-s − 2.68e6·67-s + 3.70e6·71-s − 1.53e5·73-s − 5.56e6·77-s + 7.57e6·79-s − 9.34e6·83-s + ⋯ |
| L(s) = 1 | + 0.375·5-s + 1.03·7-s − 1.34·11-s + 0.00858·13-s − 0.266·17-s + 1.61·19-s + 0.0110·23-s − 0.858·25-s − 0.958·29-s + 0.972·31-s + 0.387·35-s − 1.34·37-s − 1.42·41-s − 1.09·43-s − 0.756·47-s + 0.0660·49-s + 0.328·53-s − 0.505·55-s + 1.84·59-s + 1.51·61-s + 0.00322·65-s − 1.08·67-s + 1.22·71-s − 0.0460·73-s − 1.39·77-s + 1.72·79-s − 1.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 21 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 937 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5943 T + p^{7} T^{2} \) |
| 13 | \( 1 - 68 T + p^{7} T^{2} \) |
| 17 | \( 1 + 5400 T + p^{7} T^{2} \) |
| 19 | \( 1 - 48382 T + p^{7} T^{2} \) |
| 23 | \( 1 - 642 T + p^{7} T^{2} \) |
| 29 | \( 1 + 125934 T + p^{7} T^{2} \) |
| 31 | \( 1 - 161275 T + p^{7} T^{2} \) |
| 37 | \( 1 + 414286 T + p^{7} T^{2} \) |
| 41 | \( 1 + 627474 T + p^{7} T^{2} \) |
| 43 | \( 1 + 570590 T + p^{7} T^{2} \) |
| 47 | \( 1 + 538698 T + p^{7} T^{2} \) |
| 53 | \( 1 - 356283 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2910828 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2684168 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2681078 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3705480 T + p^{7} T^{2} \) |
| 73 | \( 1 + 153151 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7579288 T + p^{7} T^{2} \) |
| 83 | \( 1 + 9345999 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4033602 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5754097 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.699860202046557028144738878320, −8.436504709910537250510978442381, −7.83096744655816582099546349053, −6.84069472311562674480651235952, −5.37154838792834861140760737711, −5.09401830159092639641648023447, −3.58657560531639340094664074117, −2.36371878924226439339278806208, −1.40104304513926193449603459801, 0,
1.40104304513926193449603459801, 2.36371878924226439339278806208, 3.58657560531639340094664074117, 5.09401830159092639641648023447, 5.37154838792834861140760737711, 6.84069472311562674480651235952, 7.83096744655816582099546349053, 8.436504709910537250510978442381, 9.699860202046557028144738878320
