| L(s) = 1 | − 3.11·3-s + 5-s + 4.75·7-s + 6.70·9-s − 2.94·11-s + 0.885·13-s − 3.11·15-s − 1.47·17-s − 2.22·19-s − 14.8·21-s − 3.11·23-s + 25-s − 11.5·27-s − 29-s − 4.18·31-s + 9.17·33-s + 4.75·35-s − 7.51·37-s − 2.75·39-s − 0.945·41-s − 3.70·43-s + 6.70·45-s − 3.17·47-s + 15.6·49-s + 4.58·51-s + 13.9·53-s − 2.94·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 0.447·5-s + 1.79·7-s + 2.23·9-s − 0.888·11-s + 0.245·13-s − 0.804·15-s − 0.357·17-s − 0.511·19-s − 3.23·21-s − 0.649·23-s + 0.200·25-s − 2.21·27-s − 0.185·29-s − 0.752·31-s + 1.59·33-s + 0.804·35-s − 1.23·37-s − 0.441·39-s − 0.147·41-s − 0.564·43-s + 0.999·45-s − 0.463·47-s + 2.23·49-s + 0.642·51-s + 1.91·53-s − 0.397·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180827810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180827810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 - 0.885T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 + 0.945T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 4.62T + 61T^{2} \) |
| 67 | \( 1 + 0.824T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−7.47691752314120918968216348500, −7.04071241343415322134589618699, −6.09215895397005197770957426040, −5.60962049609555931208141297852, −5.01940514423501809278177650958, −4.63621673675041854067697902098, −3.76321185027808865106244288211, −2.15438189738726794609489594648, −1.66887203458437798608674962233, −0.58688685318679662660595777148,
0.58688685318679662660595777148, 1.66887203458437798608674962233, 2.15438189738726794609489594648, 3.76321185027808865106244288211, 4.63621673675041854067697902098, 5.01940514423501809278177650958, 5.60962049609555931208141297852, 6.09215895397005197770957426040, 7.04071241343415322134589618699, 7.47691752314120918968216348500
