| L(s) = 1 | − 0.400·2-s − 1.83·4-s + 2.30·5-s + 7-s + 1.53·8-s − 0.924·10-s + 4.49·13-s − 0.400·14-s + 3.06·16-s + 7.46·17-s − 1.34·19-s − 4.24·20-s − 4.27·23-s + 0.315·25-s − 1.80·26-s − 1.83·28-s + 1.12·29-s − 8.11·31-s − 4.30·32-s − 2.99·34-s + 2.30·35-s + 11.0·37-s + 0.540·38-s + 3.54·40-s − 1.16·41-s + 9.95·43-s + 1.71·46-s + ⋯ |
| L(s) = 1 | − 0.283·2-s − 0.919·4-s + 1.03·5-s + 0.377·7-s + 0.544·8-s − 0.292·10-s + 1.24·13-s − 0.107·14-s + 0.765·16-s + 1.81·17-s − 0.309·19-s − 0.948·20-s − 0.890·23-s + 0.0631·25-s − 0.353·26-s − 0.347·28-s + 0.209·29-s − 1.45·31-s − 0.761·32-s − 0.513·34-s + 0.389·35-s + 1.81·37-s + 0.0876·38-s + 0.561·40-s − 0.181·41-s + 1.51·43-s + 0.252·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169825212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169825212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.400T + 2T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 5.28T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−7.77230160467180730615242548266, −7.56733565896980566351585227594, −6.17598465325703839709573543476, −5.75983032811140634954664347393, −5.30403552105926740742615769845, −4.15757282548142171978578682628, −3.76469986690225268878704106705, −2.58021318315341015309809153665, −1.54311499801220620783212989537, −0.861558336015002929193150092628,
0.861558336015002929193150092628, 1.54311499801220620783212989537, 2.58021318315341015309809153665, 3.76469986690225268878704106705, 4.15757282548142171978578682628, 5.30403552105926740742615769845, 5.75983032811140634954664347393, 6.17598465325703839709573543476, 7.56733565896980566351585227594, 7.77230160467180730615242548266
