| L(s) = 1 | + 2-s + 1.29·3-s + 4-s + 2.87·5-s + 1.29·6-s − 3.57·7-s + 8-s − 1.31·9-s + 2.87·10-s − 1.21·11-s + 1.29·12-s − 1.04·13-s − 3.57·14-s + 3.74·15-s + 16-s − 6.73·17-s − 1.31·18-s − 2.53·19-s + 2.87·20-s − 4.64·21-s − 1.21·22-s + 23-s + 1.29·24-s + 3.29·25-s − 1.04·26-s − 5.60·27-s − 3.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.749·3-s + 0.5·4-s + 1.28·5-s + 0.530·6-s − 1.35·7-s + 0.353·8-s − 0.437·9-s + 0.910·10-s − 0.365·11-s + 0.374·12-s − 0.290·13-s − 0.956·14-s + 0.965·15-s + 0.250·16-s − 1.63·17-s − 0.309·18-s − 0.580·19-s + 0.643·20-s − 1.01·21-s − 0.258·22-s + 0.208·23-s + 0.265·24-s + 0.658·25-s − 0.205·26-s − 1.07·27-s − 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 4.03T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 + 0.722T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 8.17T + 79T^{2} \) |
| 83 | \( 1 + 2.57T + 83T^{2} \) |
| 89 | \( 1 + 8.39T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−7.52125677630814684953949421006, −6.87777457940642444729711147127, −6.07903629677925470691022506603, −5.83090989258758071056736472264, −4.82841850829156931714226136066, −3.93268391738850743144917692847, −3.06150979382753129551591099148, −2.47506762666952231817456295745, −1.89724085104450806554613771574, 0,
1.89724085104450806554613771574, 2.47506762666952231817456295745, 3.06150979382753129551591099148, 3.93268391738850743144917692847, 4.82841850829156931714226136066, 5.83090989258758071056736472264, 6.07903629677925470691022506603, 6.87777457940642444729711147127, 7.52125677630814684953949421006
