| L(s) = 1 | + 5.29·2-s − 9·3-s − 3.99·4-s + 108.·5-s − 47.6·6-s + 212.·7-s − 190.·8-s + 81·9-s + 571.·10-s − 73.5·11-s + 35.9·12-s − 431.·13-s + 1.12e3·14-s − 972.·15-s − 880.·16-s + 170.·17-s + 428.·18-s + 2.02e3·19-s − 431.·20-s − 1.90e3·21-s − 389.·22-s + 835.·23-s + 1.71e3·24-s + 8.55e3·25-s − 2.28e3·26-s − 729·27-s − 847.·28-s + ⋯ |
| L(s) = 1 | + 0.935·2-s − 0.577·3-s − 0.124·4-s + 1.93·5-s − 0.540·6-s + 1.63·7-s − 1.05·8-s + 0.333·9-s + 1.80·10-s − 0.183·11-s + 0.0720·12-s − 0.707·13-s + 1.53·14-s − 1.11·15-s − 0.859·16-s + 0.143·17-s + 0.311·18-s + 1.28·19-s − 0.241·20-s − 0.944·21-s − 0.171·22-s + 0.329·23-s + 0.607·24-s + 2.73·25-s − 0.661·26-s − 0.192·27-s − 0.204·28-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
| \(L(3)\) |
\(\approx\) |
\(3.802719094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.802719094\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 5.29T + 32T^{2} \) |
| 5 | \( 1 - 108.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 212.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 73.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 431.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 170.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 835.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 272.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.67e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.19e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 3.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
| 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
−11.94855426839054641408536016095, −10.95993294099421208779360914550, −9.827921349604585643753805947680, −9.010103734913905272512387708076, −7.41851387628109437238959894220, −5.87170057841823465000233349389, −5.36546848204829723792644669583, −4.58395776103531260505966941146, −2.55176692878869075136933253748, −1.26620540903332943478637594112,
1.26620540903332943478637594112, 2.55176692878869075136933253748, 4.58395776103531260505966941146, 5.36546848204829723792644669583, 5.87170057841823465000233349389, 7.41851387628109437238959894220, 9.010103734913905272512387708076, 9.827921349604585643753805947680, 10.95993294099421208779360914550, 11.94855426839054641408536016095
