| L(s) = 1 | − 0.394·2-s − 2.83·3-s − 1.84·4-s + 1.12·6-s + 0.914·7-s + 1.51·8-s + 5.05·9-s + 1.45·11-s + 5.23·12-s − 4.92·13-s − 0.361·14-s + 3.08·16-s − 2.55·17-s − 1.99·18-s + 2.57·19-s − 2.59·21-s − 0.572·22-s + 1.67·23-s − 4.30·24-s + 1.94·26-s − 5.83·27-s − 1.68·28-s − 4.06·29-s + 1.90·31-s − 4.25·32-s − 4.11·33-s + 1.00·34-s + ⋯ |
| L(s) = 1 | − 0.279·2-s − 1.63·3-s − 0.922·4-s + 0.457·6-s + 0.345·7-s + 0.536·8-s + 1.68·9-s + 0.437·11-s + 1.51·12-s − 1.36·13-s − 0.0965·14-s + 0.772·16-s − 0.619·17-s − 0.470·18-s + 0.591·19-s − 0.566·21-s − 0.122·22-s + 0.348·23-s − 0.879·24-s + 0.381·26-s − 1.12·27-s − 0.318·28-s − 0.753·29-s + 0.342·31-s − 0.752·32-s − 0.717·33-s + 0.172·34-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 | \( 1 + 0.394T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 7 | \( 1 - 0.914T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 1.40T + 47T^{2} \) |
| 53 | \( 1 + 9.75T + 53T^{2} \) |
| 59 | \( 1 - 6.52T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 1.14T + 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 - 8.07T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 0.985T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 0.323T + 97T^{2} \) |
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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
−7.68389100838353621021154357762, −6.89219453444204635851749414832, −6.33818437764494976083646532885, −5.25623817721117483930500968553, −5.04305854151546718527341849423, −4.42292605662653164928954538457, −3.45744606121779576993278321044, −1.97099590043887786561823869690, −0.903234951534291865560585342070, 0,
0.903234951534291865560585342070, 1.97099590043887786561823869690, 3.45744606121779576993278321044, 4.42292605662653164928954538457, 5.04305854151546718527341849423, 5.25623817721117483930500968553, 6.33818437764494976083646532885, 6.89219453444204635851749414832, 7.68389100838353621021154357762
