| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 3.85·5-s − 2.23·7-s + 2.23·8-s + 2.38·10-s + 1.38·11-s − 0.236·13-s + 1.38·14-s + 1.85·16-s + 4.38·17-s − 4.85·19-s + 6.23·20-s − 0.854·22-s + 1.23·23-s + 9.85·25-s + 0.145·26-s + 3.61·28-s − 10.0·31-s − 5.61·32-s − 2.70·34-s + 8.61·35-s + 4.70·37-s + 3.00·38-s − 8.61·40-s − 3.85·41-s − 7.23·43-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 1.72·5-s − 0.845·7-s + 0.790·8-s + 0.753·10-s + 0.416·11-s − 0.0654·13-s + 0.369·14-s + 0.463·16-s + 1.06·17-s − 1.11·19-s + 1.39·20-s − 0.182·22-s + 0.257·23-s + 1.97·25-s + 0.0286·26-s + 0.683·28-s − 1.81·31-s − 0.993·32-s − 0.464·34-s + 1.45·35-s + 0.774·37-s + 0.486·38-s − 1.36·40-s − 0.601·41-s − 1.10·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2709623026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2709623026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 3.85T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 3.56T + 97T^{2} \) |
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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
−7.83537674822379891727120988647, −7.44017077797015206223541996258, −6.72017767568210285323956542016, −5.78410466072182894272959827141, −4.88793211929119153623588473287, −4.13267674993182048909767688163, −3.68208429220805701304412300669, −3.00317137124081724324657105516, −1.43571576194055332557475879467, −0.29756994223083770618652804036,
0.29756994223083770618652804036, 1.43571576194055332557475879467, 3.00317137124081724324657105516, 3.68208429220805701304412300669, 4.13267674993182048909767688163, 4.88793211929119153623588473287, 5.78410466072182894272959827141, 6.72017767568210285323956542016, 7.44017077797015206223541996258, 7.83537674822379891727120988647
