| L(s) = 1 | − 21.5·2-s + 335.·4-s − 54.7·5-s − 565.·7-s − 4.47e3·8-s + 1.17e3·10-s + 890.·11-s + 1.24e4·13-s + 1.21e4·14-s + 5.34e4·16-s + 2.69e3·17-s − 5.60e4·19-s − 1.83e4·20-s − 1.91e4·22-s − 1.21e4·23-s − 7.51e4·25-s − 2.67e5·26-s − 1.89e5·28-s + 7.26e4·29-s + 2.07e5·31-s − 5.77e5·32-s − 5.79e4·34-s + 3.09e4·35-s − 1.60e5·37-s + 1.20e6·38-s + 2.45e5·40-s + 5.62e5·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s − 0.195·5-s − 0.622·7-s − 3.09·8-s + 0.372·10-s + 0.201·11-s + 1.56·13-s + 1.18·14-s + 3.26·16-s + 0.132·17-s − 1.87·19-s − 0.513·20-s − 0.383·22-s − 0.208·23-s − 0.961·25-s − 2.98·26-s − 1.63·28-s + 0.552·29-s + 1.24·31-s − 3.11·32-s − 0.253·34-s + 0.121·35-s − 0.520·37-s + 3.57·38-s + 0.605·40-s + 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 21.5T + 128T^{2} \) |
| 5 | \( 1 + 54.7T + 7.81e4T^{2} \) |
| 7 | \( 1 + 565.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 890.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.24e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.69e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.60e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 7.26e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.60e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.55e3T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.63e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.29e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.20e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.81e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.49e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.07e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.57e6T + 8.07e13T^{2} \) |
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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
−10.46255520955164901963605914105, −9.574881470343133392137858002129, −8.582756827117697454879289270979, −8.051183783422357342460841585702, −6.64140672728581993817964977157, −6.15467793240654228016207530277, −3.77297566301972806141274517150, −2.36323515726178655151294277520, −1.12231922570370743603659464590, 0,
1.12231922570370743603659464590, 2.36323515726178655151294277520, 3.77297566301972806141274517150, 6.15467793240654228016207530277, 6.64140672728581993817964977157, 8.051183783422357342460841585702, 8.582756827117697454879289270979, 9.574881470343133392137858002129, 10.46255520955164901963605914105
