L(s) = 1 | − 0.161·2-s − 1.97·4-s − 2.08·5-s + 3.66·7-s + 0.641·8-s + 0.335·10-s − 3.77·11-s + 3.51·13-s − 0.592·14-s + 3.84·16-s − 3.02·17-s − 1.97·19-s + 4.10·20-s + 0.609·22-s + 23-s − 0.669·25-s − 0.566·26-s − 7.24·28-s − 29-s − 3.58·31-s − 1.90·32-s + 0.488·34-s − 7.63·35-s + 5.34·37-s + 0.318·38-s − 1.33·40-s + 12.0·41-s + ⋯ |
L(s) = 1 | − 0.114·2-s − 0.986·4-s − 0.930·5-s + 1.38·7-s + 0.226·8-s + 0.106·10-s − 1.13·11-s + 0.973·13-s − 0.158·14-s + 0.961·16-s − 0.733·17-s − 0.453·19-s + 0.918·20-s + 0.130·22-s + 0.208·23-s − 0.133·25-s − 0.111·26-s − 1.36·28-s − 0.185·29-s − 0.643·31-s − 0.336·32-s + 0.0837·34-s − 1.29·35-s + 0.878·37-s + 0.0517·38-s − 0.211·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 0.161T + 2T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 3.68T + 43T^{2} \) |
| 47 | \( 1 - 0.138T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 3.28T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 + 5.82T + 83T^{2} \) |
| 89 | \( 1 + 6.06T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75326783050617329829955420241, −7.52078791863272783245059518642, −6.14768637801084131679208013051, −5.45530089619803093070510781278, −4.56985746077452200771931595298, −4.32301350655711589936875213048, −3.42386088399910151596128410449, −2.26301968237805607199296893701, −1.13017656210983374175216924077, 0,
1.13017656210983374175216924077, 2.26301968237805607199296893701, 3.42386088399910151596128410449, 4.32301350655711589936875213048, 4.56985746077452200771931595298, 5.45530089619803093070510781278, 6.14768637801084131679208013051, 7.52078791863272783245059518642, 7.75326783050617329829955420241