| L(s) = 1 | − 63.7·2-s + 2.01e3·4-s + 3.12e3·5-s + 4.19e4·7-s + 2.20e3·8-s − 1.99e5·10-s + 9.57e5·11-s + 1.39e6·13-s − 2.67e6·14-s − 4.26e6·16-s − 3.76e6·17-s + 9.41e6·19-s + 6.29e6·20-s − 6.10e7·22-s − 3.02e7·23-s + 9.76e6·25-s − 8.86e7·26-s + 8.44e7·28-s − 1.03e8·29-s − 5.48e7·31-s + 2.67e8·32-s + 2.40e8·34-s + 1.31e8·35-s + 4.78e8·37-s − 5.99e8·38-s + 6.88e6·40-s + 9.29e8·41-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.983·4-s + 0.447·5-s + 0.942·7-s + 0.0237·8-s − 0.629·10-s + 1.79·11-s + 1.03·13-s − 1.32·14-s − 1.01·16-s − 0.643·17-s + 0.871·19-s + 0.439·20-s − 2.52·22-s − 0.981·23-s + 0.199·25-s − 1.46·26-s + 0.926·28-s − 0.937·29-s − 0.344·31-s + 1.40·32-s + 0.906·34-s + 0.421·35-s + 1.13·37-s − 1.22·38-s + 0.0106·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.396436142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396436142\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3.12e3T \) |
| good | 2 | \( 1 + 63.7T + 2.04e3T^{2} \) |
| 7 | \( 1 - 4.19e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.57e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.39e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.76e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.41e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.02e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.48e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.78e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.29e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.68e7T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.20e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.02e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.97e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.07e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.61e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 6.64e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.57e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.04e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.21e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.10e11T + 7.15e21T^{2} \) |
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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
−13.59106970106290456839370614740, −11.69758221571725753929748401754, −10.90304658380817722644706725155, −9.483558352005119969532702819207, −8.773033566273907711214682152651, −7.52842368174729526223788513249, −6.12918124445120266732680931286, −4.16813968501628161083361706858, −1.83037914699779268756640265577, −0.986321171235488763153926762507,
0.986321171235488763153926762507, 1.83037914699779268756640265577, 4.16813968501628161083361706858, 6.12918124445120266732680931286, 7.52842368174729526223788513249, 8.773033566273907711214682152651, 9.483558352005119969532702819207, 10.90304658380817722644706725155, 11.69758221571725753929748401754, 13.59106970106290456839370614740
