| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 4·13-s − 3·14-s − 15-s + 16-s + 6·17-s − 18-s + 3·19-s − 20-s + 3·21-s + 22-s − 24-s + 25-s − 4·26-s + 27-s + 3·28-s − 9·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.654·21-s + 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.566·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.61018538873747, −12.99648310350234, −12.31565069485351, −11.87552684452529, −11.54750054841971, −10.97647766748728, −10.65759900853852, −9.964885611891688, −9.720721616982493, −8.916476105395233, −8.670404815308286, −8.059605190766378, −7.824776883170211, −7.402801582662584, −6.896112162219057, −6.084212495547839, −5.596661210481780, −5.145135501600455, −4.394791785473093, −3.873550798977631, −3.224390994819857, −2.897767543503940, −1.950891093121629, −1.410302262381624, −1.060587240352466, 0,
1.060587240352466, 1.410302262381624, 1.950891093121629, 2.897767543503940, 3.224390994819857, 3.873550798977631, 4.394791785473093, 5.145135501600455, 5.596661210481780, 6.084212495547839, 6.896112162219057, 7.402801582662584, 7.824776883170211, 8.059605190766378, 8.670404815308286, 8.916476105395233, 9.720721616982493, 9.964885611891688, 10.65759900853852, 10.97647766748728, 11.54750054841971, 11.87552684452529, 12.31565069485351, 12.99648310350234, 13.61018538873747
