| L(s) = 1 | − 2-s + 0.245·3-s + 4-s + 5-s − 0.245·6-s + 7-s − 8-s − 2.93·9-s − 10-s + 0.245·12-s − 1.15·13-s − 14-s + 0.245·15-s + 16-s + 6.55·17-s + 2.93·18-s − 3.43·19-s + 20-s + 0.245·21-s + 8.30·23-s − 0.245·24-s + 25-s + 1.15·26-s − 1.46·27-s + 28-s − 2.93·29-s − 0.245·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.141·3-s + 0.5·4-s + 0.447·5-s − 0.100·6-s + 0.377·7-s − 0.353·8-s − 0.979·9-s − 0.316·10-s + 0.0709·12-s − 0.319·13-s − 0.267·14-s + 0.0634·15-s + 0.250·16-s + 1.58·17-s + 0.692·18-s − 0.787·19-s + 0.223·20-s + 0.0536·21-s + 1.73·23-s − 0.0501·24-s + 0.200·25-s + 0.225·26-s − 0.281·27-s + 0.188·28-s − 0.544·29-s − 0.0448·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.245T + 3T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 6.17T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 9.98T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 0.380T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−7.45673876221718815206291222095, −6.96516946337854723777581903882, −6.06988456490246593436445796001, −5.39913527272419352003673241941, −4.91993122347368307572108638300, −3.54425770186855121775385120354, −3.02749877023445003085097525901, −2.06486235148566586663901449192, −1.27815601544182813310722691760, 0,
1.27815601544182813310722691760, 2.06486235148566586663901449192, 3.02749877023445003085097525901, 3.54425770186855121775385120354, 4.91993122347368307572108638300, 5.39913527272419352003673241941, 6.06988456490246593436445796001, 6.96516946337854723777581903882, 7.45673876221718815206291222095
