| L(s) = 1 | − 3.62e6·2-s + 4.35e12·4-s − 6.19e14·5-s + 2.58e18·7-s + 1.60e19·8-s + 2.24e21·10-s − 2.74e22·11-s + 6.62e23·13-s − 9.38e24·14-s − 9.67e25·16-s − 5.82e25·17-s − 1.98e27·19-s − 2.70e27·20-s + 9.94e28·22-s + 2.61e29·23-s − 7.52e29·25-s − 2.40e30·26-s + 1.12e31·28-s − 2.27e31·29-s − 1.73e32·31-s + 2.09e32·32-s + 2.11e32·34-s − 1.60e33·35-s + 5.95e32·37-s + 7.18e33·38-s − 9.97e33·40-s + 5.56e34·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.495·4-s − 0.581·5-s + 1.75·7-s + 0.616·8-s + 0.710·10-s − 1.11·11-s + 0.743·13-s − 2.14·14-s − 1.24·16-s − 0.204·17-s − 0.636·19-s − 0.288·20-s + 1.36·22-s + 1.37·23-s − 0.662·25-s − 0.909·26-s + 0.867·28-s − 0.824·29-s − 1.49·31-s + 0.911·32-s + 0.249·34-s − 1.01·35-s + 0.114·37-s + 0.777·38-s − 0.358·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
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 | \( 1 + 3.62e6T + 8.79e12T^{2} \) |
| 5 | \( 1 + 6.19e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 2.58e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 2.74e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 6.62e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 5.82e25T + 8.11e52T^{2} \) |
| 19 | \( 1 + 1.98e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 2.61e29T + 3.58e58T^{2} \) |
| 29 | \( 1 + 2.27e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.73e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 5.95e32T + 2.70e67T^{2} \) |
| 41 | \( 1 - 5.56e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 1.13e35T + 1.73e70T^{2} \) |
| 47 | \( 1 + 2.48e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 3.93e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 6.06e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 3.31e38T + 5.87e76T^{2} \) |
| 67 | \( 1 - 2.46e38T + 3.32e78T^{2} \) |
| 71 | \( 1 - 5.22e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.32e40T + 1.32e80T^{2} \) |
| 79 | \( 1 + 1.99e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 5.56e39T + 3.31e82T^{2} \) |
| 89 | \( 1 - 2.64e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 3.46e42T + 2.69e85T^{2} \) |
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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
−11.19950633007055225766543971451, −10.80093623694837808359325321068, −9.031370507917707667918998926930, −8.080601081576979107041630167110, −7.42463618505245359191442290439, −5.28503995053206949188933294855, −4.14439920341881441545562709712, −2.17504734060588388529396012817, −1.15417591742006860674466290076, 0,
1.15417591742006860674466290076, 2.17504734060588388529396012817, 4.14439920341881441545562709712, 5.28503995053206949188933294855, 7.42463618505245359191442290439, 8.080601081576979107041630167110, 9.031370507917707667918998926930, 10.80093623694837808359325321068, 11.19950633007055225766543971451
