| L(s) = 1 | − 2.32·2-s + 0.941·3-s + 3.40·4-s − 2.38·5-s − 2.18·6-s + 2.74·7-s − 3.26·8-s − 2.11·9-s + 5.53·10-s − 1.86·11-s + 3.20·12-s − 0.255·13-s − 6.38·14-s − 2.24·15-s + 0.775·16-s − 1.20·17-s + 4.91·18-s + 5.01·19-s − 8.10·20-s + 2.58·21-s + 4.32·22-s − 7.51·23-s − 3.07·24-s + 0.677·25-s + 0.593·26-s − 4.81·27-s + 9.34·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 0.543·3-s + 1.70·4-s − 1.06·5-s − 0.893·6-s + 1.03·7-s − 1.15·8-s − 0.704·9-s + 1.75·10-s − 0.561·11-s + 0.925·12-s − 0.0708·13-s − 1.70·14-s − 0.579·15-s + 0.193·16-s − 0.291·17-s + 1.15·18-s + 1.15·19-s − 1.81·20-s + 0.564·21-s + 0.922·22-s − 1.56·23-s − 0.627·24-s + 0.135·25-s + 0.116·26-s − 0.926·27-s + 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 0.941T + 3T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 0.255T + 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 - 2.71T + 47T^{2} \) |
| 53 | \( 1 - 5.72T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 8.99T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + 4.49T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.04T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−10.12243917310531481546768480498, −8.996859783812358312367100238980, −8.473240287166241596717857612060, −7.66574861312230153355419383351, −7.45804484245243782481532197632, −5.81207077534201767947658140883, −4.43349893559890047905380959400, −3.03021347320900409801281676469, −1.76122918587429918489294910481, 0,
1.76122918587429918489294910481, 3.03021347320900409801281676469, 4.43349893559890047905380959400, 5.81207077534201767947658140883, 7.45804484245243782481532197632, 7.66574861312230153355419383351, 8.473240287166241596717857612060, 8.996859783812358312367100238980, 10.12243917310531481546768480498
