| L(s) = 1 | − 9.14·2-s − 44.4·4-s + 27.9·5-s − 1.41e3·7-s + 1.57e3·8-s − 255.·10-s + 4.88e3·11-s − 5.21e3·13-s + 1.29e4·14-s − 8.71e3·16-s − 5.79e3·17-s + 2.49e4·19-s − 1.24e3·20-s − 4.46e4·22-s + 6.30e4·23-s − 7.73e4·25-s + 4.76e4·26-s + 6.29e4·28-s − 8.65e4·29-s + 2.68e5·31-s − 1.22e5·32-s + 5.29e4·34-s − 3.96e4·35-s − 2.35e5·37-s − 2.27e5·38-s + 4.40e4·40-s + 5.85e5·41-s + ⋯ |
| L(s) = 1 | − 0.807·2-s − 0.347·4-s + 0.100·5-s − 1.56·7-s + 1.08·8-s − 0.0808·10-s + 1.10·11-s − 0.657·13-s + 1.26·14-s − 0.532·16-s − 0.286·17-s + 0.834·19-s − 0.0347·20-s − 0.894·22-s + 1.07·23-s − 0.989·25-s + 0.531·26-s + 0.541·28-s − 0.659·29-s + 1.61·31-s − 0.658·32-s + 0.231·34-s − 0.156·35-s − 0.763·37-s − 0.673·38-s + 0.108·40-s + 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6657784365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6657784365\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 9.14T + 128T^{2} \) |
| 5 | \( 1 - 27.9T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.41e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.79e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.65e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.68e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.35e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.57e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.07e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.98e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.17e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.38e6T + 8.07e13T^{2} \) |
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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
−9.537990233961415161618730180227, −9.188428685272188923813154077252, −8.045859389166220596591514159871, −7.02577716194826467266927375372, −6.34189839428313622253059659878, −5.05644998468810320162783068764, −3.93417139772052341737302798067, −2.96732304685785968994328625502, −1.49013964299579342890635310887, −0.42337111925351864380158580169,
0.42337111925351864380158580169, 1.49013964299579342890635310887, 2.96732304685785968994328625502, 3.93417139772052341737302798067, 5.05644998468810320162783068764, 6.34189839428313622253059659878, 7.02577716194826467266927375372, 8.045859389166220596591514159871, 9.188428685272188923813154077252, 9.537990233961415161618730180227
