| L(s) = 1 | + 2.56·2-s − 9·3-s − 25.4·4-s − 23.1·6-s + 82.6·7-s − 147.·8-s + 81·9-s + 483.·11-s + 228.·12-s + 7.50·13-s + 212.·14-s + 434.·16-s + 1.08e3·17-s + 207.·18-s + 3.04e3·19-s − 743.·21-s + 1.24e3·22-s − 3.38e3·23-s + 1.32e3·24-s + 19.2·26-s − 729·27-s − 2.09e3·28-s + 2.34e3·29-s + 3.91e3·31-s + 5.83e3·32-s − 4.35e3·33-s + 2.77e3·34-s + ⋯ |
| L(s) = 1 | + 0.453·2-s − 0.577·3-s − 0.793·4-s − 0.262·6-s + 0.637·7-s − 0.814·8-s + 0.333·9-s + 1.20·11-s + 0.458·12-s + 0.0123·13-s + 0.289·14-s + 0.424·16-s + 0.906·17-s + 0.151·18-s + 1.93·19-s − 0.367·21-s + 0.547·22-s − 1.33·23-s + 0.470·24-s + 0.00559·26-s − 0.192·27-s − 0.506·28-s + 0.517·29-s + 0.731·31-s + 1.00·32-s − 0.696·33-s + 0.411·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.676884444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676884444\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.56T + 32T^{2} \) |
| 7 | \( 1 - 82.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 483.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 7.50T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 3.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.34e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.69e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.82e4T + 8.58e9T^{2} \) |
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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
−13.87116178366219855943605077291, −12.18044059166172331914661079117, −11.81911021568275022796946700605, −10.16576743182755240288151917012, −9.105384672710450502470991472878, −7.69671035119999400620446984594, −6.03260866154167651105957285519, −4.94419692492687297011832114783, −3.66799091153197237441082371214, −1.04977524744511636072588525309,
1.04977524744511636072588525309, 3.66799091153197237441082371214, 4.94419692492687297011832114783, 6.03260866154167651105957285519, 7.69671035119999400620446984594, 9.105384672710450502470991472878, 10.16576743182755240288151917012, 11.81911021568275022796946700605, 12.18044059166172331914661079117, 13.87116178366219855943605077291
