L(s) = 1 | − 2-s + 2.17·3-s + 4-s − 1.56·5-s − 2.17·6-s + 0.0344·7-s − 8-s + 1.71·9-s + 1.56·10-s − 3.93·11-s + 2.17·12-s − 5.20·13-s − 0.0344·14-s − 3.40·15-s + 16-s + 4.85·17-s − 1.71·18-s − 7.17·19-s − 1.56·20-s + 0.0746·21-s + 3.93·22-s − 23-s − 2.17·24-s − 2.53·25-s + 5.20·26-s − 2.79·27-s + 0.0344·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.25·3-s + 0.5·4-s − 0.701·5-s − 0.886·6-s + 0.0130·7-s − 0.353·8-s + 0.570·9-s + 0.496·10-s − 1.18·11-s + 0.626·12-s − 1.44·13-s − 0.00919·14-s − 0.879·15-s + 0.250·16-s + 1.17·17-s − 0.403·18-s − 1.64·19-s − 0.350·20-s + 0.0162·21-s + 0.839·22-s − 0.208·23-s − 0.443·24-s − 0.507·25-s + 1.02·26-s − 0.537·27-s + 0.00650·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244883942\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244883942\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 0.0344T + 7T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 - 0.916T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 2.21T + 53T^{2} \) |
| 59 | \( 1 - 7.55T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 5.56T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.112578099704720318659577335003, −7.69069490627857138606726485217, −7.10887769370783019180885181792, −6.07463419170043441176074047383, −5.15021171487182053520700453917, −4.24726054849892218788449742077, −3.44314955714250742463994429045, −2.46020934988214295934729519436, −2.27429434983142794902480100710, −0.57313641616713741685113080798,
0.57313641616713741685113080798, 2.27429434983142794902480100710, 2.46020934988214295934729519436, 3.44314955714250742463994429045, 4.24726054849892218788449742077, 5.15021171487182053520700453917, 6.07463419170043441176074047383, 7.10887769370783019180885181792, 7.69069490627857138606726485217, 8.112578099704720318659577335003