| L(s) = 1 | − 0.188·2-s − 1.96·4-s + 1.33·5-s − 0.912·7-s + 0.748·8-s − 0.253·10-s + 11-s + 5.78·13-s + 0.172·14-s + 3.78·16-s − 3.08·17-s − 6.38·19-s − 2.63·20-s − 0.188·22-s − 7.04·23-s − 3.20·25-s − 1.09·26-s + 1.79·28-s + 9.45·29-s + 7.73·31-s − 2.21·32-s + 0.583·34-s − 1.22·35-s − 8.78·37-s + 1.20·38-s + 1.00·40-s − 8.00·41-s + ⋯ |
| L(s) = 1 | − 0.133·2-s − 0.982·4-s + 0.599·5-s − 0.345·7-s + 0.264·8-s − 0.0800·10-s + 0.301·11-s + 1.60·13-s + 0.0460·14-s + 0.946·16-s − 0.749·17-s − 1.46·19-s − 0.588·20-s − 0.0402·22-s − 1.46·23-s − 0.640·25-s − 0.214·26-s + 0.338·28-s + 1.75·29-s + 1.38·31-s − 0.391·32-s + 0.100·34-s − 0.206·35-s − 1.44·37-s + 0.195·38-s + 0.158·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.188T + 2T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 7 | \( 1 + 0.912T + 7T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 - 9.45T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 + 8.00T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 8.28T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−8.172743681017892869805513919034, −6.73472241185343688928926046549, −6.30369693522891112325685915853, −5.72940474261381308855843342882, −4.66712908963527669099820409784, −4.11491689736668747428006131804, −3.39798396534633040854775486750, −2.19435779009705395993388244808, −1.26390083974005360040316246041, 0,
1.26390083974005360040316246041, 2.19435779009705395993388244808, 3.39798396534633040854775486750, 4.11491689736668747428006131804, 4.66712908963527669099820409784, 5.72940474261381308855843342882, 6.30369693522891112325685915853, 6.73472241185343688928926046549, 8.172743681017892869805513919034
