| L(s) = 1 | + 2-s + 2.96·3-s + 4-s − 3.00·5-s + 2.96·6-s − 7-s + 8-s + 5.79·9-s − 3.00·10-s − 1.37·11-s + 2.96·12-s + 0.937·13-s − 14-s − 8.91·15-s + 16-s − 3.48·17-s + 5.79·18-s − 6.44·19-s − 3.00·20-s − 2.96·21-s − 1.37·22-s − 4.32·23-s + 2.96·24-s + 4.03·25-s + 0.937·26-s + 8.30·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s − 1.34·5-s + 1.21·6-s − 0.377·7-s + 0.353·8-s + 1.93·9-s − 0.950·10-s − 0.414·11-s + 0.856·12-s + 0.260·13-s − 0.267·14-s − 2.30·15-s + 0.250·16-s − 0.844·17-s + 1.36·18-s − 1.47·19-s − 0.672·20-s − 0.647·21-s − 0.292·22-s − 0.901·23-s + 0.605·24-s + 0.807·25-s + 0.183·26-s + 1.59·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 0.937T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 - 0.0370T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 0.915T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 0.176T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 0.855T + 73T^{2} \) |
| 79 | \( 1 - 8.20T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−7.77565190457155329927081447312, −7.10323574859997255691437379897, −6.59956252108989272688521921710, −5.40500283967444778347722723580, −4.37796952834406912867850879541, −3.81391328120309647995138453526, −3.51870385212815316811080035343, −2.47762285401849788967980993154, −1.88583338079477005463203106120, 0,
1.88583338079477005463203106120, 2.47762285401849788967980993154, 3.51870385212815316811080035343, 3.81391328120309647995138453526, 4.37796952834406912867850879541, 5.40500283967444778347722723580, 6.59956252108989272688521921710, 7.10323574859997255691437379897, 7.77565190457155329927081447312
