| L(s) = 1 | + 1.73·2-s + 3-s + 1.00·4-s − 1.94·5-s + 1.73·6-s + 4.25·7-s − 1.72·8-s + 9-s − 3.37·10-s − 4.96·11-s + 1.00·12-s − 5.19·13-s + 7.38·14-s − 1.94·15-s − 4.99·16-s + 1.81·17-s + 1.73·18-s − 7.12·19-s − 1.95·20-s + 4.25·21-s − 8.61·22-s − 1.40·23-s − 1.72·24-s − 1.20·25-s − 9.00·26-s + 27-s + 4.28·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.503·4-s − 0.870·5-s + 0.707·6-s + 1.60·7-s − 0.609·8-s + 0.333·9-s − 1.06·10-s − 1.49·11-s + 0.290·12-s − 1.44·13-s + 1.97·14-s − 0.502·15-s − 1.24·16-s + 0.441·17-s + 0.408·18-s − 1.63·19-s − 0.438·20-s + 0.929·21-s − 1.83·22-s − 0.292·23-s − 0.351·24-s − 0.241·25-s − 1.76·26-s + 0.192·27-s + 0.810·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 37 | \( 1 + 3.57T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 0.877T + 43T^{2} \) |
| 47 | \( 1 - 2.86T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.141022125788695777209573238363, −7.74457979220548143614011841856, −7.02772869944014481224023299132, −5.67648974565733122494525815512, −5.07386116323263181902926569652, −4.43826464938239667037389268782, −3.87583916618471186973243597327, −2.66051372930993468982013852195, −2.10219555706588060812662217980, 0,
2.10219555706588060812662217980, 2.66051372930993468982013852195, 3.87583916618471186973243597327, 4.43826464938239667037389268782, 5.07386116323263181902926569652, 5.67648974565733122494525815512, 7.02772869944014481224023299132, 7.74457979220548143614011841856, 8.141022125788695777209573238363
