| L(s) = 1 | + 2-s + 4-s − 2.22·5-s − 0.652·7-s + 8-s − 2.22·10-s + 1.53·11-s − 0.467·13-s − 0.652·14-s + 16-s + 2.10·17-s − 2.22·20-s + 1.53·22-s − 5.90·23-s − 0.0418·25-s − 0.467·26-s − 0.652·28-s + 8.33·29-s − 8.63·31-s + 32-s + 2.10·34-s + 1.45·35-s − 4.67·37-s − 2.22·40-s − 3.47·41-s + 10.2·43-s + 1.53·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.995·5-s − 0.246·7-s + 0.353·8-s − 0.704·10-s + 0.461·11-s − 0.129·13-s − 0.174·14-s + 0.250·16-s + 0.510·17-s − 0.497·20-s + 0.326·22-s − 1.23·23-s − 0.00837·25-s − 0.0917·26-s − 0.123·28-s + 1.54·29-s − 1.55·31-s + 0.176·32-s + 0.361·34-s + 0.245·35-s − 0.768·37-s − 0.352·40-s − 0.542·41-s + 1.56·43-s + 0.230·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 0.467T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 - 8.33T + 29T^{2} \) |
| 31 | \( 1 + 8.63T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.63T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 0.248T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
| 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
−7.57099787260880088820306804830, −6.98069352778777831480687209429, −6.17012492517355626515794206796, −5.54527781138102432830950477135, −4.61111318954675544538739396306, −3.97038237252079385062695620542, −3.43757706134649204716297324342, −2.51472912326475955676500852292, −1.38954129519612776904434526801, 0,
1.38954129519612776904434526801, 2.51472912326475955676500852292, 3.43757706134649204716297324342, 3.97038237252079385062695620542, 4.61111318954675544538739396306, 5.54527781138102432830950477135, 6.17012492517355626515794206796, 6.98069352778777831480687209429, 7.57099787260880088820306804830
