| L(s) = 1 | − 2-s − 4-s + 5-s + 7-s + 3·8-s − 10-s + 3·11-s + 7·13-s − 14-s − 16-s + 8·19-s − 20-s − 3·22-s − 4·23-s + 25-s − 7·26-s − 28-s − 2·29-s − 5·31-s − 5·32-s + 35-s + 2·37-s − 8·38-s + 3·40-s − 3·41-s + 4·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s − 0.316·10-s + 0.904·11-s + 1.94·13-s − 0.267·14-s − 1/4·16-s + 1.83·19-s − 0.223·20-s − 0.639·22-s − 0.834·23-s + 1/5·25-s − 1.37·26-s − 0.188·28-s − 0.371·29-s − 0.898·31-s − 0.883·32-s + 0.169·35-s + 0.328·37-s − 1.29·38-s + 0.474·40-s − 0.468·41-s + 0.609·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.99619591128986, −13.64250235578839, −13.33748711253753, −12.65939731484604, −12.08228178208391, −11.46171732326688, −11.03666441049311, −10.71235581311308, −9.917046697924873, −9.532191845052451, −9.179476790761808, −8.679128165382878, −8.156249311763297, −7.690922607397097, −7.175125944404750, −6.371403536969613, −5.976092611097932, −5.413959115308058, −4.776421577926481, −4.161753534972389, −3.552310386698793, −3.171290272363188, −1.934246445566049, −1.351013785831544, −1.140328309457336, 0,
1.140328309457336, 1.351013785831544, 1.934246445566049, 3.171290272363188, 3.552310386698793, 4.161753534972389, 4.776421577926481, 5.413959115308058, 5.976092611097932, 6.371403536969613, 7.175125944404750, 7.690922607397097, 8.156249311763297, 8.679128165382878, 9.179476790761808, 9.532191845052451, 9.917046697924873, 10.71235581311308, 11.03666441049311, 11.46171732326688, 12.08228178208391, 12.65939731484604, 13.33748711253753, 13.64250235578839, 13.99619591128986
