| L(s) = 1 | + 2.66·2-s + 0.955·3-s + 5.12·4-s + 1.71·5-s + 2.55·6-s − 0.402·7-s + 8.34·8-s − 2.08·9-s + 4.56·10-s − 2.27·11-s + 4.89·12-s − 6.19·13-s − 1.07·14-s + 1.63·15-s + 12.0·16-s + 17-s − 5.56·18-s + 6.85·19-s + 8.77·20-s − 0.384·21-s − 6.05·22-s + 3.93·23-s + 7.97·24-s − 2.07·25-s − 16.5·26-s − 4.86·27-s − 2.06·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.551·3-s + 2.56·4-s + 0.765·5-s + 1.04·6-s − 0.151·7-s + 2.95·8-s − 0.695·9-s + 1.44·10-s − 0.684·11-s + 1.41·12-s − 1.71·13-s − 0.286·14-s + 0.422·15-s + 3.00·16-s + 0.242·17-s − 1.31·18-s + 1.57·19-s + 1.96·20-s − 0.0838·21-s − 1.29·22-s + 0.820·23-s + 1.62·24-s − 0.414·25-s − 3.24·26-s − 0.935·27-s − 0.389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.937258637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.937258637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.955T + 3T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 + 0.402T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 - 3.93T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 5.55T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 - 8.91T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
| 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
−9.971309360093438350193437315905, −9.432306728346690901843489969849, −7.84737050218601300104839754694, −7.31774487831824086173613987835, −6.24324409709935982616302591364, −5.27515199335589548955344439103, −5.05577985081209106877494176400, −3.53661530982233947159515447600, −2.80525381073005141091151701636, −2.06522921530377426840386140215,
2.06522921530377426840386140215, 2.80525381073005141091151701636, 3.53661530982233947159515447600, 5.05577985081209106877494176400, 5.27515199335589548955344439103, 6.24324409709935982616302591364, 7.31774487831824086173613987835, 7.84737050218601300104839754694, 9.432306728346690901843489969849, 9.971309360093438350193437315905
