| L(s) = 1 | − 0.540·3-s − 2·4-s + 3.35·5-s + 0.459·7-s − 2.70·9-s + 4.89·11-s + 1.08·12-s + 13-s − 1.81·15-s + 4·16-s + 6.70·17-s − 2.89·19-s − 6.70·20-s − 0.248·21-s − 23-s + 6.24·25-s + 3.08·27-s − 0.918·28-s + 9.78·29-s − 4·31-s − 2.64·33-s + 1.54·35-s + 5.41·36-s − 3.16·37-s − 0.540·39-s − 3.62·41-s − 7.62·43-s + ⋯ |
| L(s) = 1 | − 0.312·3-s − 4-s + 1.49·5-s + 0.173·7-s − 0.902·9-s + 1.47·11-s + 0.312·12-s + 0.277·13-s − 0.468·15-s + 16-s + 1.62·17-s − 0.664·19-s − 1.49·20-s − 0.0541·21-s − 0.208·23-s + 1.24·25-s + 0.593·27-s − 0.173·28-s + 1.81·29-s − 0.718·31-s − 0.460·33-s + 0.260·35-s + 0.902·36-s − 0.520·37-s − 0.0865·39-s − 0.566·41-s − 1.16·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.262626379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262626379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 0.540T + 3T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 - 0.459T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 29 | \( 1 - 9.78T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + 3.62T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 + 8.91T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 4.98T + 97T^{2} \) |
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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
−11.91143495508773611599182473414, −10.59922595973722181595886849827, −9.782262535001237216918960323947, −9.041192239600045072152945876797, −8.236536290424497760254220670730, −6.46029751017406852981076493878, −5.78515651959489484283134976983, −4.82976358620818444189439612054, −3.32901461767990072080076612376, −1.39792459294861940749715711325,
1.39792459294861940749715711325, 3.32901461767990072080076612376, 4.82976358620818444189439612054, 5.78515651959489484283134976983, 6.46029751017406852981076493878, 8.236536290424497760254220670730, 9.041192239600045072152945876797, 9.782262535001237216918960323947, 10.59922595973722181595886849827, 11.91143495508773611599182473414
