| L(s) = 1 | − 3.05·3-s + 27.7·5-s + 103.·7-s − 233.·9-s + 158.·11-s − 578.·13-s − 84.7·15-s + 253.·17-s + 3.09e3·19-s − 317.·21-s − 4.16e3·23-s − 2.35e3·25-s + 1.45e3·27-s + 6.77e3·29-s − 6.26e3·31-s − 484.·33-s + 2.87e3·35-s − 3.29e3·37-s + 1.77e3·39-s − 6.15e3·41-s + 1.84e3·43-s − 6.47e3·45-s − 8.15e3·47-s − 6.05e3·49-s − 776.·51-s − 3.04e4·53-s + 4.38e3·55-s + ⋯ |
| L(s) = 1 | − 0.196·3-s + 0.495·5-s + 0.799·7-s − 0.961·9-s + 0.394·11-s − 0.950·13-s − 0.0972·15-s + 0.213·17-s + 1.96·19-s − 0.156·21-s − 1.64·23-s − 0.754·25-s + 0.384·27-s + 1.49·29-s − 1.17·31-s − 0.0774·33-s + 0.396·35-s − 0.395·37-s + 0.186·39-s − 0.571·41-s + 0.152·43-s − 0.476·45-s − 0.538·47-s − 0.360·49-s − 0.0418·51-s − 1.48·53-s + 0.195·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 3 | \( 1 + 3.05T + 243T^{2} \) |
| 5 | \( 1 - 27.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 158.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 578.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 253.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 3.09e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.15e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 8.15e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.25e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.94e4T + 8.58e9T^{2} \) |
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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
−9.463025411276367335278228368422, −8.290215524887779101092537532362, −7.67151285577936716924184312392, −6.52192296928151237804506757012, −5.51765693211709914990207965794, −4.97925334737494941785535255722, −3.60114764404426453520576059772, −2.43054357564583299476877726383, −1.37474867488948856245032918495, 0,
1.37474867488948856245032918495, 2.43054357564583299476877726383, 3.60114764404426453520576059772, 4.97925334737494941785535255722, 5.51765693211709914990207965794, 6.52192296928151237804506757012, 7.67151285577936716924184312392, 8.290215524887779101092537532362, 9.463025411276367335278228368422
