| L(s) = 1 | + 2.57·2-s + 1.18·3-s + 4.64·4-s + 1.23·5-s + 3.04·6-s + 0.678·7-s + 6.82·8-s − 1.60·9-s + 3.19·10-s + 1.13·11-s + 5.48·12-s − 13-s + 1.74·14-s + 1.46·15-s + 8.29·16-s − 4.20·17-s − 4.13·18-s − 0.201·19-s + 5.75·20-s + 0.801·21-s + 2.92·22-s + 2.94·23-s + 8.05·24-s − 3.46·25-s − 2.57·26-s − 5.43·27-s + 3.15·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.681·3-s + 2.32·4-s + 0.554·5-s + 1.24·6-s + 0.256·7-s + 2.41·8-s − 0.535·9-s + 1.01·10-s + 0.342·11-s + 1.58·12-s − 0.277·13-s + 0.467·14-s + 0.377·15-s + 2.07·16-s − 1.02·17-s − 0.975·18-s − 0.0461·19-s + 1.28·20-s + 0.174·21-s + 0.624·22-s + 0.613·23-s + 1.64·24-s − 0.693·25-s − 0.505·26-s − 1.04·27-s + 0.595·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.473086477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.473086477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 0.678T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 + 0.201T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.40T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 0.684T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−9.577397253967837381898294849431, −8.820121885683871651136930627962, −7.72404918455178106470077207126, −6.92344103441164648363889617084, −5.99844982801171296376330772525, −5.41896034487647339943962833469, −4.40790384110787964208878955457, −3.62953328837058534769487254864, −2.60200332912402604123245396898, −1.94654023012865123987259933643,
1.94654023012865123987259933643, 2.60200332912402604123245396898, 3.62953328837058534769487254864, 4.40790384110787964208878955457, 5.41896034487647339943962833469, 5.99844982801171296376330772525, 6.92344103441164648363889617084, 7.72404918455178106470077207126, 8.820121885683871651136930627962, 9.577397253967837381898294849431
