| L(s) = 1 | + 2-s + 0.385·3-s + 4-s − 0.181·5-s + 0.385·6-s + 4.50·7-s + 8-s − 2.85·9-s − 0.181·10-s − 11-s + 0.385·12-s + 0.761·13-s + 4.50·14-s − 0.0700·15-s + 16-s − 6.46·17-s − 2.85·18-s − 0.181·20-s + 1.73·21-s − 22-s − 3.57·23-s + 0.385·24-s − 4.96·25-s + 0.761·26-s − 2.25·27-s + 4.50·28-s − 2.59·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.222·3-s + 0.5·4-s − 0.0813·5-s + 0.157·6-s + 1.70·7-s + 0.353·8-s − 0.950·9-s − 0.0575·10-s − 0.301·11-s + 0.111·12-s + 0.211·13-s + 1.20·14-s − 0.0180·15-s + 0.250·16-s − 1.56·17-s − 0.672·18-s − 0.0406·20-s + 0.378·21-s − 0.213·22-s − 0.746·23-s + 0.0786·24-s − 0.993·25-s + 0.149·26-s − 0.433·27-s + 0.850·28-s − 0.482·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.385T + 3T^{2} \) |
| 5 | \( 1 + 0.181T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 13 | \( 1 - 0.761T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.03T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 + 0.995T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 2.63T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−7.60676692068315683687136280060, −6.77815565884750920503237970674, −5.85620409446693575679431234536, −5.39943264759823758950622415057, −4.62889701622997284534431153520, −4.05883211423796976776922419317, −3.17225916068366999525321273566, −2.08780269100083208724401032297, −1.78262817768820759926666798913, 0,
1.78262817768820759926666798913, 2.08780269100083208724401032297, 3.17225916068366999525321273566, 4.05883211423796976776922419317, 4.62889701622997284534431153520, 5.39943264759823758950622415057, 5.85620409446693575679431234536, 6.77815565884750920503237970674, 7.60676692068315683687136280060
