| L(s) = 1 | − 2-s + 3.02·3-s + 4-s − 3.36·5-s − 3.02·6-s + 7-s − 8-s + 6.17·9-s + 3.36·10-s + 5.02·11-s + 3.02·12-s − 4.83·13-s − 14-s − 10.2·15-s + 16-s − 0.146·17-s − 6.17·18-s − 3.36·20-s + 3.02·21-s − 5.02·22-s + 4.56·23-s − 3.02·24-s + 6.34·25-s + 4.83·26-s + 9.62·27-s + 28-s + 7.52·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.74·3-s + 0.5·4-s − 1.50·5-s − 1.23·6-s + 0.377·7-s − 0.353·8-s + 2.05·9-s + 1.06·10-s + 1.51·11-s + 0.874·12-s − 1.34·13-s − 0.267·14-s − 2.63·15-s + 0.250·16-s − 0.0355·17-s − 1.45·18-s − 0.753·20-s + 0.661·21-s − 1.07·22-s + 0.951·23-s − 0.618·24-s + 1.26·25-s + 0.948·26-s + 1.85·27-s + 0.188·28-s + 1.39·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.315004069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315004069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 17 | \( 1 + 0.146T + 17T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 - 7.52T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 + 0.970T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 0.882T + 67T^{2} \) |
| 71 | \( 1 - 1.29T + 71T^{2} \) |
| 73 | \( 1 - 9.85T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.256512909062363572333522411752, −7.78752580379480996066101627884, −6.98905947046548626796037598876, −6.78978048071038382966895211939, −4.98178933434660021615833928048, −4.23598146350894138557169621817, −3.57201394816716796797612050087, −2.88560640298570422928950727207, −1.95299902440275788035689500859, −0.869873322731514697339235470150,
0.869873322731514697339235470150, 1.95299902440275788035689500859, 2.88560640298570422928950727207, 3.57201394816716796797612050087, 4.23598146350894138557169621817, 4.98178933434660021615833928048, 6.78978048071038382966895211939, 6.98905947046548626796037598876, 7.78752580379480996066101627884, 8.256512909062363572333522411752
