L(s) = 1 | + 2-s + 4-s − 1.25·5-s + 3.30·7-s + 8-s − 1.25·10-s + 3.08·11-s + 1.64·13-s + 3.30·14-s + 16-s + 7.25·17-s + 5.93·19-s − 1.25·20-s + 3.08·22-s + 9.00·23-s − 3.42·25-s + 1.64·26-s + 3.30·28-s + 4.86·29-s − 10.1·31-s + 32-s + 7.25·34-s − 4.15·35-s − 2.70·37-s + 5.93·38-s − 1.25·40-s − 6.19·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.561·5-s + 1.25·7-s + 0.353·8-s − 0.396·10-s + 0.929·11-s + 0.455·13-s + 0.884·14-s + 0.250·16-s + 1.76·17-s + 1.36·19-s − 0.280·20-s + 0.657·22-s + 1.87·23-s − 0.685·25-s + 0.321·26-s + 0.625·28-s + 0.904·29-s − 1.83·31-s + 0.176·32-s + 1.24·34-s − 0.701·35-s − 0.444·37-s + 0.963·38-s − 0.198·40-s − 0.967·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.322817582\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.322817582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
good | 5 | \( 1 + 1.25T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 9.00T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.75835270775699332551824365793, −7.44679808646647146566834890065, −6.65438495437864312243862498521, −5.63293672731375450161341350176, −5.16921144422721880727385262085, −4.49107858363304168461013792968, −3.46856779107790488765138200832, −3.22357546946906989969894996798, −1.64948105201285502933733551258, −1.13036807828125403008250031969,
1.13036807828125403008250031969, 1.64948105201285502933733551258, 3.22357546946906989969894996798, 3.46856779107790488765138200832, 4.49107858363304168461013792968, 5.16921144422721880727385262085, 5.63293672731375450161341350176, 6.65438495437864312243862498521, 7.44679808646647146566834890065, 7.75835270775699332551824365793