L(s) = 1 | − 0.478·3-s − 5-s + 1.59·7-s − 2.77·9-s − 6.51·11-s − 3.37·13-s + 0.478·15-s − 3.23·17-s − 0.00948·19-s − 0.762·21-s + 8.46·23-s + 25-s + 2.76·27-s + 0.667·29-s − 2.17·31-s + 3.12·33-s − 1.59·35-s − 9.25·37-s + 1.61·39-s − 5.04·41-s − 5.75·43-s + 2.77·45-s + 3.87·47-s − 4.46·49-s + 1.55·51-s + 3.05·53-s + 6.51·55-s + ⋯ |
L(s) = 1 | − 0.276·3-s − 0.447·5-s + 0.601·7-s − 0.923·9-s − 1.96·11-s − 0.936·13-s + 0.123·15-s − 0.785·17-s − 0.00217·19-s − 0.166·21-s + 1.76·23-s + 0.200·25-s + 0.531·27-s + 0.124·29-s − 0.391·31-s + 0.543·33-s − 0.269·35-s − 1.52·37-s + 0.258·39-s − 0.788·41-s − 0.877·43-s + 0.413·45-s + 0.564·47-s − 0.638·49-s + 0.217·51-s + 0.419·53-s + 0.878·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5453752561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5453752561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.478T + 3T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 6.51T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 + 0.00948T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 - 0.667T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 9.25T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 - 3.04T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 1.98T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 - 3.02T + 97T^{2} \) |
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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
−7.77017384930784439827503441180, −7.27823957778641915129624719867, −6.53149961059532871496080310698, −5.46421245418607756283535713435, −5.03816592099572931428677400024, −4.65374662155682173342834866039, −3.26788735639239268022848703661, −2.77514983034011894650864431468, −1.87592970799625197033736683393, −0.34803621964698552120335681630,
0.34803621964698552120335681630, 1.87592970799625197033736683393, 2.77514983034011894650864431468, 3.26788735639239268022848703661, 4.65374662155682173342834866039, 5.03816592099572931428677400024, 5.46421245418607756283535713435, 6.53149961059532871496080310698, 7.27823957778641915129624719867, 7.77017384930784439827503441180