| L(s) = 1 | + 2.25·2-s + 0.826·3-s + 3.06·4-s + 1.85·6-s − 0.537·7-s + 2.39·8-s − 2.31·9-s − 6.19·11-s + 2.53·12-s + 4.65·13-s − 1.20·14-s − 0.735·16-s − 3.65·17-s − 5.21·18-s + 3.14·19-s − 0.443·21-s − 13.9·22-s − 1.86·23-s + 1.98·24-s + 10.4·26-s − 4.39·27-s − 1.64·28-s − 4.37·29-s − 3.31·31-s − 6.44·32-s − 5.11·33-s − 8.22·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.476·3-s + 1.53·4-s + 0.759·6-s − 0.203·7-s + 0.847·8-s − 0.772·9-s − 1.86·11-s + 0.730·12-s + 1.29·13-s − 0.323·14-s − 0.183·16-s − 0.886·17-s − 1.22·18-s + 0.721·19-s − 0.0968·21-s − 2.97·22-s − 0.388·23-s + 0.404·24-s + 2.05·26-s − 0.845·27-s − 0.311·28-s − 0.812·29-s − 0.595·31-s − 1.14·32-s − 0.890·33-s − 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 3 | \( 1 - 0.826T + 3T^{2} \) |
| 7 | \( 1 + 0.537T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
| 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
−7.65160051235060878541651876149, −6.82064299105299210740516171425, −5.89052755757361186463368180154, −5.63307039039020810048967866891, −4.84002315055311529239156772010, −3.99225132320461075583309399675, −3.22711354207906714810085278177, −2.74854370599748798125307156123, −1.90225342397831756214580883419, 0,
1.90225342397831756214580883419, 2.74854370599748798125307156123, 3.22711354207906714810085278177, 3.99225132320461075583309399675, 4.84002315055311529239156772010, 5.63307039039020810048967866891, 5.89052755757361186463368180154, 6.82064299105299210740516171425, 7.65160051235060878541651876149
