| L(s) = 1 | − 2.58·2-s − 1.31·4-s + 5·5-s − 7·7-s + 24.0·8-s − 12.9·10-s − 38.2·11-s + 19.3·13-s + 18.1·14-s − 51.7·16-s + 87.2·17-s − 44.2·19-s − 6.56·20-s + 98.9·22-s − 218.·23-s + 25·25-s − 50.0·26-s + 9.19·28-s + 46.9·29-s + 194.·31-s − 58.8·32-s − 225.·34-s − 35·35-s + 366.·37-s + 114.·38-s + 120.·40-s + 339.·41-s + ⋯ |
| L(s) = 1 | − 0.914·2-s − 0.164·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.408·10-s − 1.04·11-s + 0.412·13-s + 0.345·14-s − 0.808·16-s + 1.24·17-s − 0.534·19-s − 0.0734·20-s + 0.958·22-s − 1.97·23-s + 0.200·25-s − 0.377·26-s + 0.0620·28-s + 0.300·29-s + 1.12·31-s − 0.324·32-s − 1.13·34-s − 0.169·35-s + 1.63·37-s + 0.488·38-s + 0.475·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9131149391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9131149391\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.58T + 8T^{2} \) |
| 11 | \( 1 + 38.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 218.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 226.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 11.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 - 320.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 14.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 952T + 3.57e5T^{2} \) |
| 73 | \( 1 - 824.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 156.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 170.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−10.82595620169439374386044625293, −10.03506708339436406126858493542, −9.589197403687613012345096762524, −8.250511204401108437830195045069, −7.85796112771084375656852881012, −6.39462802812522710275969801258, −5.33990507519871392978892684932, −4.00330066673777330629276434001, −2.35611731956218467381565749202, −0.75815921534852244250859828078,
0.75815921534852244250859828078, 2.35611731956218467381565749202, 4.00330066673777330629276434001, 5.33990507519871392978892684932, 6.39462802812522710275969801258, 7.85796112771084375656852881012, 8.250511204401108437830195045069, 9.589197403687613012345096762524, 10.03506708339436406126858493542, 10.82595620169439374386044625293
