| L(s) = 1 | + 1.22·2-s − 3-s − 0.494·4-s − 4.26·5-s − 1.22·6-s − 7-s − 3.06·8-s + 9-s − 5.22·10-s + 0.264·11-s + 0.494·12-s + 2.88·13-s − 1.22·14-s + 4.26·15-s − 2.76·16-s − 3.86·17-s + 1.22·18-s − 5.28·19-s + 2.10·20-s + 21-s + 0.324·22-s + 6.29·23-s + 3.06·24-s + 13.1·25-s + 3.54·26-s − 27-s + 0.494·28-s + ⋯ |
| L(s) = 1 | + 0.867·2-s − 0.577·3-s − 0.247·4-s − 1.90·5-s − 0.500·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s − 1.65·10-s + 0.0798·11-s + 0.142·12-s + 0.801·13-s − 0.327·14-s + 1.10·15-s − 0.691·16-s − 0.938·17-s + 0.289·18-s − 1.21·19-s + 0.471·20-s + 0.218·21-s + 0.0692·22-s + 1.31·23-s + 0.624·24-s + 2.63·25-s + 0.694·26-s − 0.192·27-s + 0.0935·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 11 | \( 1 - 0.264T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 - 0.0252T + 31T^{2} \) |
| 37 | \( 1 - 7.69T + 37T^{2} \) |
| 41 | \( 1 - 0.969T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 - 7.95T + 53T^{2} \) |
| 59 | \( 1 - 4.31T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 - 7.91T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 1.16T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 3.13T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−7.30509545953771052108466011974, −6.61477144377520928969890134632, −6.19754054151158270992743613271, −5.09013575004617580111305592806, −4.57337817128102932206676827771, −3.99920521599176483577276917557, −3.49171185848439215742771359585, −2.62596239793366171731145675201, −0.874196938400001458553386411088, 0,
0.874196938400001458553386411088, 2.62596239793366171731145675201, 3.49171185848439215742771359585, 3.99920521599176483577276917557, 4.57337817128102932206676827771, 5.09013575004617580111305592806, 6.19754054151158270992743613271, 6.61477144377520928969890134632, 7.30509545953771052108466011974
