L(s) = 1 | + 2-s − 3.04·3-s + 4-s + 5-s − 3.04·6-s − 0.894·7-s + 8-s + 6.26·9-s + 10-s + 6.48·11-s − 3.04·12-s + 3.45·13-s − 0.894·14-s − 3.04·15-s + 16-s + 4.19·17-s + 6.26·18-s + 0.973·19-s + 20-s + 2.72·21-s + 6.48·22-s + 1.93·23-s − 3.04·24-s + 25-s + 3.45·26-s − 9.95·27-s − 0.894·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s + 0.447·5-s − 1.24·6-s − 0.338·7-s + 0.353·8-s + 2.08·9-s + 0.316·10-s + 1.95·11-s − 0.878·12-s + 0.957·13-s − 0.239·14-s − 0.786·15-s + 0.250·16-s + 1.01·17-s + 1.47·18-s + 0.223·19-s + 0.223·20-s + 0.594·21-s + 1.38·22-s + 0.404·23-s − 0.621·24-s + 0.200·25-s + 0.676·26-s − 1.91·27-s − 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.329028078\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.329028078\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.894T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 - 3.45T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 - 0.973T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 0.364T + 53T^{2} \) |
| 59 | \( 1 - 0.324T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 7.29T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + 4.49T + 97T^{2} \) |
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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
−8.423862217881181767761137376217, −7.22288482728647881971786490554, −6.46828869068333687201106414414, −6.33906555130799156189693870620, −5.53063772028570431771275208994, −4.87586706685443375940425695733, −3.98633453661090711665397474545, −3.31690423510983625106809495628, −1.59174765677052604800600213659, −0.977142982424957899433946784593,
0.977142982424957899433946784593, 1.59174765677052604800600213659, 3.31690423510983625106809495628, 3.98633453661090711665397474545, 4.87586706685443375940425695733, 5.53063772028570431771275208994, 6.33906555130799156189693870620, 6.46828869068333687201106414414, 7.22288482728647881971786490554, 8.423862217881181767761137376217