| L(s) = 1 | − 2.21·2-s − 2.76·3-s + 2.91·4-s + 5-s + 6.13·6-s − 3.75·7-s − 2.02·8-s + 4.65·9-s − 2.21·10-s − 1.54·11-s − 8.06·12-s + 4.15·13-s + 8.31·14-s − 2.76·15-s − 1.33·16-s + 5.74·17-s − 10.3·18-s + 7.03·19-s + 2.91·20-s + 10.3·21-s + 3.41·22-s − 6.77·23-s + 5.60·24-s + 25-s − 9.20·26-s − 4.57·27-s − 10.9·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 1.59·3-s + 1.45·4-s + 0.447·5-s + 2.50·6-s − 1.41·7-s − 0.716·8-s + 1.55·9-s − 0.700·10-s − 0.464·11-s − 2.32·12-s + 1.15·13-s + 2.22·14-s − 0.714·15-s − 0.334·16-s + 1.39·17-s − 2.43·18-s + 1.61·19-s + 0.651·20-s + 2.26·21-s + 0.728·22-s − 1.41·23-s + 1.14·24-s + 0.200·25-s − 1.80·26-s − 0.880·27-s − 2.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 + 2.76T + 3T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 0.0578T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 8.83T + 79T^{2} \) |
| 83 | \( 1 + 9.09T + 83T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
| 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
−10.25167788269709518976945104052, −10.03476433586651522582004550837, −9.193675616170500042491329003522, −7.86044296999292340328126928018, −6.95781374742630329202725193015, −6.05078860209947002756367539400, −5.44832777227399333657953202112, −3.45587790298020185856735092222, −1.38471060990701707155474720672, 0,
1.38471060990701707155474720672, 3.45587790298020185856735092222, 5.44832777227399333657953202112, 6.05078860209947002756367539400, 6.95781374742630329202725193015, 7.86044296999292340328126928018, 9.193675616170500042491329003522, 10.03476433586651522582004550837, 10.25167788269709518976945104052
