| L(s) = 1 | − 9.80e8·2-s − 1.34e18·4-s − 1.59e20·5-s + 1.63e25·7-s + 3.57e27·8-s + 1.56e29·10-s − 1.92e30·11-s − 1.30e34·13-s − 1.60e34·14-s − 4.08e35·16-s − 2.98e37·17-s − 7.88e38·19-s + 2.14e38·20-s + 1.88e39·22-s + 4.50e41·23-s − 4.31e42·25-s + 1.28e43·26-s − 2.19e43·28-s + 5.51e44·29-s + 4.17e45·31-s − 7.85e45·32-s + 2.93e46·34-s − 2.60e45·35-s − 6.95e47·37-s + 7.72e47·38-s − 5.70e47·40-s − 1.25e49·41-s + ⋯ |
| L(s) = 1 | − 0.645·2-s − 0.583·4-s − 0.0765·5-s + 0.274·7-s + 1.02·8-s + 0.0494·10-s − 0.0332·11-s − 1.38·13-s − 0.177·14-s − 0.0769·16-s − 0.884·17-s − 0.784·19-s + 0.0446·20-s + 0.0214·22-s + 1.31·23-s − 0.994·25-s + 0.894·26-s − 0.159·28-s + 1.37·29-s + 1.36·31-s − 0.972·32-s + 0.571·34-s − 0.0210·35-s − 1.02·37-s + 0.506·38-s − 0.0782·40-s − 0.809·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(62-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+61/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(31)\) |
\(\approx\) |
\(0.6320370140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6320370140\) |
| \(L(\frac{63}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 9.80e8T + 2.30e18T^{2} \) |
| 5 | \( 1 + 1.59e20T + 4.33e42T^{2} \) |
| 7 | \( 1 - 1.63e25T + 3.55e51T^{2} \) |
| 11 | \( 1 + 1.92e30T + 3.34e63T^{2} \) |
| 13 | \( 1 + 1.30e34T + 8.92e67T^{2} \) |
| 17 | \( 1 + 2.98e37T + 1.14e75T^{2} \) |
| 19 | \( 1 + 7.88e38T + 1.00e78T^{2} \) |
| 23 | \( 1 - 4.50e41T + 1.16e83T^{2} \) |
| 29 | \( 1 - 5.51e44T + 1.60e89T^{2} \) |
| 31 | \( 1 - 4.17e45T + 9.39e90T^{2} \) |
| 37 | \( 1 + 6.95e47T + 4.57e95T^{2} \) |
| 41 | \( 1 + 1.25e49T + 2.39e98T^{2} \) |
| 43 | \( 1 - 8.92e49T + 4.38e99T^{2} \) |
| 47 | \( 1 + 1.21e50T + 9.95e101T^{2} \) |
| 53 | \( 1 + 5.55e52T + 1.51e105T^{2} \) |
| 59 | \( 1 + 4.29e53T + 1.05e108T^{2} \) |
| 61 | \( 1 - 1.17e53T + 8.03e108T^{2} \) |
| 67 | \( 1 + 3.47e55T + 2.45e111T^{2} \) |
| 71 | \( 1 + 3.07e56T + 8.44e112T^{2} \) |
| 73 | \( 1 - 7.42e56T + 4.59e113T^{2} \) |
| 79 | \( 1 + 4.19e57T + 5.69e115T^{2} \) |
| 83 | \( 1 + 3.40e58T + 1.15e117T^{2} \) |
| 89 | \( 1 - 3.77e59T + 8.18e118T^{2} \) |
| 97 | \( 1 + 2.29e60T + 1.55e121T^{2} \) |
| 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
−10.78206033044030961727234202278, −9.794228599629500667745619547323, −8.771905708916704185128184474450, −7.82498888085043893500221645532, −6.67455944207189323102087623567, −4.99507104154431916061494709669, −4.36201135786314025798649917807, −2.75804243209388297577030510564, −1.57122493427827703868463562425, −0.37298418887931835903153852410,
0.37298418887931835903153852410, 1.57122493427827703868463562425, 2.75804243209388297577030510564, 4.36201135786314025798649917807, 4.99507104154431916061494709669, 6.67455944207189323102087623567, 7.82498888085043893500221645532, 8.771905708916704185128184474450, 9.794228599629500667745619547323, 10.78206033044030961727234202278
