| L(s) = 1 | − 5.17·2-s − 5.19·4-s − 25·5-s − 123.·7-s + 192.·8-s + 129.·10-s + 121·11-s − 500.·13-s + 639.·14-s − 830.·16-s + 422.·17-s − 932.·19-s + 129.·20-s − 626.·22-s + 1.22e3·23-s + 625·25-s + 2.58e3·26-s + 641.·28-s + 2.11e3·29-s − 159.·31-s − 1.86e3·32-s − 2.18e3·34-s + 3.08e3·35-s + 5.41e3·37-s + 4.82e3·38-s − 4.81e3·40-s + 1.80e4·41-s + ⋯ |
| L(s) = 1 | − 0.915·2-s − 0.162·4-s − 0.447·5-s − 0.952·7-s + 1.06·8-s + 0.409·10-s + 0.301·11-s − 0.820·13-s + 0.871·14-s − 0.811·16-s + 0.354·17-s − 0.592·19-s + 0.0726·20-s − 0.275·22-s + 0.482·23-s + 0.200·25-s + 0.751·26-s + 0.154·28-s + 0.466·29-s − 0.0298·31-s − 0.321·32-s − 0.324·34-s + 0.425·35-s + 0.650·37-s + 0.542·38-s − 0.475·40-s + 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 5.17T + 32T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 500.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 422.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 932.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 159.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.81e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.15e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.15e4T + 8.58e9T^{2} \) |
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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
−9.550330587405674172816570520770, −9.010395154214905927803105305614, −7.945724898565198369811142546381, −7.23630259922108459755315303122, −6.20505484078671843287063878252, −4.82999171935734890545137962615, −3.86965023704842386845184881482, −2.56401086506499728258623315751, −0.976516323932939592350109957027, 0,
0.976516323932939592350109957027, 2.56401086506499728258623315751, 3.86965023704842386845184881482, 4.82999171935734890545137962615, 6.20505484078671843287063878252, 7.23630259922108459755315303122, 7.945724898565198369811142546381, 9.010395154214905927803105305614, 9.550330587405674172816570520770
