| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 2·14-s − 15-s + 16-s + 17-s − 18-s + 2·19-s + 20-s − 2·21-s + 22-s − 3·23-s + 24-s + 25-s − 27-s + 2·28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.13106489594005, −13.70183560894537, −13.19578476561548, −12.45076416468264, −12.13187481818464, −11.56944954326914, −11.20403726683414, −10.57876225711377, −10.21345074640515, −9.754788913755682, −9.260391340633209, −8.514994965818668, −8.136541142786941, −7.728483650308826, −6.886822737248374, −6.704060531564630, −5.881201614913843, −5.485236271627970, −4.896642942521034, −4.403764604712476, −3.491567628772660, −2.906418775568593, −2.046669331919912, −1.592029990771539, −0.8911367067023357, 0,
0.8911367067023357, 1.592029990771539, 2.046669331919912, 2.906418775568593, 3.491567628772660, 4.403764604712476, 4.896642942521034, 5.485236271627970, 5.881201614913843, 6.704060531564630, 6.886822737248374, 7.728483650308826, 8.136541142786941, 8.514994965818668, 9.260391340633209, 9.754788913755682, 10.21345074640515, 10.57876225711377, 11.20403726683414, 11.56944954326914, 12.13187481818464, 12.45076416468264, 13.19578476561548, 13.70183560894537, 14.13106489594005
