| L(s) = 1 | + 7.93·2-s + 24.4·3-s + 30.9·4-s − 25·5-s + 194.·6-s − 71.9·7-s − 8.40·8-s + 355.·9-s − 198.·10-s + 121·11-s + 756.·12-s − 338.·13-s − 570.·14-s − 611.·15-s − 1.05e3·16-s − 308.·17-s + 2.82e3·18-s + 361·19-s − 773.·20-s − 1.75e3·21-s + 959.·22-s − 3.59e3·23-s − 205.·24-s + 625·25-s − 2.68e3·26-s + 2.75e3·27-s − 2.22e3·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 1.56·3-s + 0.966·4-s − 0.447·5-s + 2.20·6-s − 0.554·7-s − 0.0464·8-s + 1.46·9-s − 0.627·10-s + 0.301·11-s + 1.51·12-s − 0.555·13-s − 0.777·14-s − 0.701·15-s − 1.03·16-s − 0.258·17-s + 2.05·18-s + 0.229·19-s − 0.432·20-s − 0.870·21-s + 0.422·22-s − 1.41·23-s − 0.0728·24-s + 0.200·25-s − 0.778·26-s + 0.726·27-s − 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 7.93T + 32T^{2} \) |
| 3 | \( 1 - 24.4T + 243T^{2} \) |
| 7 | \( 1 + 71.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 338.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 308.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 849.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.71T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.62e5T + 8.58e9T^{2} \) |
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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
−8.733930032299892847907045090121, −7.902873492465089342678604913587, −7.04007661574592936468732452854, −6.20057561131571683681831022443, −5.02701478640325281203794051909, −4.06111800465758981707040253547, −3.53315365787363365658034786550, −2.75849856336231454943754098787, −1.88258002690620215526444186413, 0,
1.88258002690620215526444186413, 2.75849856336231454943754098787, 3.53315365787363365658034786550, 4.06111800465758981707040253547, 5.02701478640325281203794051909, 6.20057561131571683681831022443, 7.04007661574592936468732452854, 7.902873492465089342678604913587, 8.733930032299892847907045090121
