| L(s) = 1 | + 6.91·2-s + 9·3-s + 15.7·4-s − 11.1·5-s + 62.2·6-s − 193.·7-s − 112.·8-s + 81·9-s − 76.9·10-s − 125.·11-s + 142.·12-s + 1.11e3·13-s − 1.33e3·14-s − 100.·15-s − 1.27e3·16-s − 1.94e3·17-s + 559.·18-s − 2.11e3·19-s − 175.·20-s − 1.73e3·21-s − 864.·22-s − 2.53e3·23-s − 1.00e3·24-s − 3.00e3·25-s + 7.71e3·26-s + 729·27-s − 3.05e3·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.493·4-s − 0.199·5-s + 0.705·6-s − 1.49·7-s − 0.619·8-s + 0.333·9-s − 0.243·10-s − 0.311·11-s + 0.284·12-s + 1.83·13-s − 1.82·14-s − 0.114·15-s − 1.24·16-s − 1.63·17-s + 0.407·18-s − 1.34·19-s − 0.0982·20-s − 0.860·21-s − 0.380·22-s − 1.00·23-s − 0.357·24-s − 0.960·25-s + 2.23·26-s + 0.192·27-s − 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 6.91T + 32T^{2} \) |
| 5 | \( 1 + 11.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 193.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 125.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.94e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 902.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.33e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.69e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.14e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.57840830707846919362715449457, −10.39287310371657117447852616030, −9.154077283802484816751963979125, −8.352218611031379589932367938300, −6.52054855763515670630576108535, −6.12328169831198028712228551780, −4.29053251326394826218770699554, −3.62941653775026072006100413011, −2.43163010351496519199752879567, 0,
2.43163010351496519199752879567, 3.62941653775026072006100413011, 4.29053251326394826218770699554, 6.12328169831198028712228551780, 6.52054855763515670630576108535, 8.352218611031379589932367938300, 9.154077283802484816751963979125, 10.39287310371657117447852616030, 11.57840830707846919362715449457
