| L(s) = 1 | + 1.64·2-s − 3-s + 0.699·4-s − 0.972·5-s − 1.64·6-s − 7-s − 2.13·8-s + 9-s − 1.59·10-s + 0.843·11-s − 0.699·12-s + 1.18·13-s − 1.64·14-s + 0.972·15-s − 4.90·16-s + 3.22·17-s + 1.64·18-s − 0.850·19-s − 0.680·20-s + 21-s + 1.38·22-s + 5.22·23-s + 2.13·24-s − 4.05·25-s + 1.94·26-s − 27-s − 0.699·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.349·4-s − 0.435·5-s − 0.670·6-s − 0.377·7-s − 0.755·8-s + 0.333·9-s − 0.505·10-s + 0.254·11-s − 0.201·12-s + 0.328·13-s − 0.439·14-s + 0.251·15-s − 1.22·16-s + 0.781·17-s + 0.387·18-s − 0.195·19-s − 0.152·20-s + 0.218·21-s + 0.295·22-s + 1.08·23-s + 0.436·24-s − 0.810·25-s + 0.381·26-s − 0.192·27-s − 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926855907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926855907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 5 | \( 1 + 0.972T + 5T^{2} \) |
| 11 | \( 1 - 0.843T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 0.850T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 8.39T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 + 0.861T + 43T^{2} \) |
| 47 | \( 1 + 7.59T + 47T^{2} \) |
| 53 | \( 1 - 2.06T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 - 7.72T + 79T^{2} \) |
| 83 | \( 1 + 0.948T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 0.0300T + 97T^{2} \) |
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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
−7.68338201950878815333062233870, −6.83453867170866998109643233424, −6.29440585582315944788816997435, −5.63543859585434513395271699713, −5.02909450727861925094322793629, −4.33515150377998991784693733369, −3.57077144706852866711181243750, −3.15019922238608711027650888628, −1.88816278925524632558079280127, −0.58334401941909360822386884427,
0.58334401941909360822386884427, 1.88816278925524632558079280127, 3.15019922238608711027650888628, 3.57077144706852866711181243750, 4.33515150377998991784693733369, 5.02909450727861925094322793629, 5.63543859585434513395271699713, 6.29440585582315944788816997435, 6.83453867170866998109643233424, 7.68338201950878815333062233870
