| L(s) = 1 | + 2-s + 0.683·3-s + 4-s − 1.10·5-s + 0.683·6-s + 2.08·7-s + 8-s − 2.53·9-s − 1.10·10-s − 4.58·11-s + 0.683·12-s + 5.34·13-s + 2.08·14-s − 0.757·15-s + 16-s − 5.12·17-s − 2.53·18-s − 5.86·19-s − 1.10·20-s + 1.42·21-s − 4.58·22-s + 4.77·23-s + 0.683·24-s − 3.77·25-s + 5.34·26-s − 3.78·27-s + 2.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.394·3-s + 0.5·4-s − 0.495·5-s + 0.279·6-s + 0.788·7-s + 0.353·8-s − 0.844·9-s − 0.350·10-s − 1.38·11-s + 0.197·12-s + 1.48·13-s + 0.557·14-s − 0.195·15-s + 0.250·16-s − 1.24·17-s − 0.596·18-s − 1.34·19-s − 0.247·20-s + 0.311·21-s − 0.977·22-s + 0.996·23-s + 0.139·24-s − 0.754·25-s + 1.04·26-s − 0.727·27-s + 0.394·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.683T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 0.116T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 0.747T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 7.81T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 - 7.92T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 0.730T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
| 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
−8.319754104998675051228673451124, −7.37688831520694496904755400992, −6.54506291652729030222561579747, −5.67909068014818012877850655393, −5.09255265031635451848997588956, −4.17561311056351174693006185158, −3.51970620528428882325855523663, −2.55061877444992591613759602157, −1.79581668381363058589344706490, 0,
1.79581668381363058589344706490, 2.55061877444992591613759602157, 3.51970620528428882325855523663, 4.17561311056351174693006185158, 5.09255265031635451848997588956, 5.67909068014818012877850655393, 6.54506291652729030222561579747, 7.37688831520694496904755400992, 8.319754104998675051228673451124
