| L(s) = 1 | + 2·2-s + 4·4-s − 3.94·5-s + 7·7-s + 8·8-s − 7.89·10-s + 21.9·11-s − 39.6·13-s + 14·14-s + 16·16-s + 91.5·17-s + 130.·19-s − 15.7·20-s + 43.8·22-s + 91.7·23-s − 109.·25-s − 79.3·26-s + 28·28-s − 69.1·29-s + 124.·31-s + 32·32-s + 183.·34-s − 27.6·35-s + 70.6·37-s + 260.·38-s − 31.5·40-s + 333.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.353·5-s + 0.377·7-s + 0.353·8-s − 0.249·10-s + 0.601·11-s − 0.846·13-s + 0.267·14-s + 0.250·16-s + 1.30·17-s + 1.57·19-s − 0.176·20-s + 0.425·22-s + 0.831·23-s − 0.875·25-s − 0.598·26-s + 0.188·28-s − 0.443·29-s + 0.722·31-s + 0.176·32-s + 0.924·34-s − 0.133·35-s + 0.314·37-s + 1.11·38-s − 0.124·40-s + 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.119153876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.119153876\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 3.94T + 125T^{2} \) |
| 11 | \( 1 - 21.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 91.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 70.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 333.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 592.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 39.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 125.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 505.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 919.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 953.T + 9.12e5T^{2} \) |
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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
−11.24862674510541206204580179001, −10.06749096855998357182374760178, −9.219294613433096094100638762640, −7.76666032212052653565604746646, −7.30502062175969930242733399210, −5.91048630198430873101566653431, −5.03523969108477652440598016283, −3.92178260760268105521324692553, −2.80810570904161337598584584988, −1.15282199288566372934254743179,
1.15282199288566372934254743179, 2.80810570904161337598584584988, 3.92178260760268105521324692553, 5.03523969108477652440598016283, 5.91048630198430873101566653431, 7.30502062175969930242733399210, 7.76666032212052653565604746646, 9.219294613433096094100638762640, 10.06749096855998357182374760178, 11.24862674510541206204580179001
