| L(s) = 1 | − 2.72·2-s + 13.3·3-s − 24.5·4-s − 25·5-s − 36.3·6-s − 136.·7-s + 154.·8-s − 65.8·9-s + 68.2·10-s − 121·11-s − 326.·12-s + 969.·13-s + 371.·14-s − 332.·15-s + 364.·16-s − 48.1·17-s + 179.·18-s + 361·19-s + 613.·20-s − 1.81e3·21-s + 330.·22-s − 4.54e3·23-s + 2.05e3·24-s + 625·25-s − 2.64e3·26-s − 4.11e3·27-s + 3.34e3·28-s + ⋯ |
| L(s) = 1 | − 0.482·2-s + 0.853·3-s − 0.767·4-s − 0.447·5-s − 0.411·6-s − 1.05·7-s + 0.852·8-s − 0.271·9-s + 0.215·10-s − 0.301·11-s − 0.655·12-s + 1.59·13-s + 0.506·14-s − 0.381·15-s + 0.356·16-s − 0.0404·17-s + 0.130·18-s + 0.229·19-s + 0.343·20-s − 0.896·21-s + 0.145·22-s − 1.78·23-s + 0.727·24-s + 0.200·25-s − 0.767·26-s − 1.08·27-s + 0.806·28-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
| \(L(3)\) |
\(\approx\) |
\(0.8325502967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8325502967\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.72T + 32T^{2} \) |
| 3 | \( 1 - 13.3T + 243T^{2} \) |
| 7 | \( 1 + 136.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 969.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 48.1T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.53e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.58e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.30e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.52e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e4T + 8.58e9T^{2} \) |
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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
−8.943995593837360468793905999730, −8.390758948660703726480017258465, −8.050783549363389598330279646705, −6.75293762087336408316498065945, −5.91267524521043740378947899159, −4.67054408641059471156928754245, −3.58607089226637184182844173886, −3.18755464652896500890718693475, −1.68530019035459050070245858063, −0.40895032914553841296496301967,
0.40895032914553841296496301967, 1.68530019035459050070245858063, 3.18755464652896500890718693475, 3.58607089226637184182844173886, 4.67054408641059471156928754245, 5.91267524521043740378947899159, 6.75293762087336408316498065945, 8.050783549363389598330279646705, 8.390758948660703726480017258465, 8.943995593837360468793905999730
