| L(s) = 1 | − 4.82·2-s + 17.2·3-s − 8.67·4-s − 25·5-s − 83.3·6-s + 225.·7-s + 196.·8-s + 55.0·9-s + 120.·10-s + 121·11-s − 149.·12-s + 270.·13-s − 1.09e3·14-s − 431.·15-s − 671.·16-s − 79.8·17-s − 265.·18-s − 361·19-s + 216.·20-s + 3.89e3·21-s − 584.·22-s − 3.34e3·23-s + 3.39e3·24-s + 625·25-s − 1.30e3·26-s − 3.24e3·27-s − 1.95e3·28-s + ⋯ |
| L(s) = 1 | − 0.853·2-s + 1.10·3-s − 0.270·4-s − 0.447·5-s − 0.945·6-s + 1.74·7-s + 1.08·8-s + 0.226·9-s + 0.381·10-s + 0.301·11-s − 0.300·12-s + 0.443·13-s − 1.48·14-s − 0.495·15-s − 0.655·16-s − 0.0669·17-s − 0.193·18-s − 0.229·19-s + 0.121·20-s + 1.92·21-s − 0.257·22-s − 1.31·23-s + 1.20·24-s + 0.200·25-s − 0.378·26-s − 0.856·27-s − 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.357355004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357355004\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 4.82T + 32T^{2} \) |
| 3 | \( 1 - 17.2T + 243T^{2} \) |
| 7 | \( 1 - 225.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 270.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 79.8T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.56e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 23.1T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.53e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.48e4T + 8.58e9T^{2} \) |
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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
−8.834779082101186224971148395571, −8.358431605769721667219301566180, −8.012779615315461758452749855726, −7.23165803648097211543937946370, −5.75835517014546773408697342285, −4.45743159061746808922138788449, −4.11161916444114793721774810230, −2.62655453562404670546914317474, −1.65380503799633529720653234349, −0.76680656621098797637424160229,
0.76680656621098797637424160229, 1.65380503799633529720653234349, 2.62655453562404670546914317474, 4.11161916444114793721774810230, 4.45743159061746808922138788449, 5.75835517014546773408697342285, 7.23165803648097211543937946370, 8.012779615315461758452749855726, 8.358431605769721667219301566180, 8.834779082101186224971148395571
