| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s − 2·13-s + 14-s + 16-s + 6·17-s − 18-s − 4·19-s + 21-s + 22-s + 6·23-s + 24-s + 2·26-s − 27-s − 28-s − 4·31-s − 32-s + 33-s − 6·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.89185000147877, −16.32684581323117, −15.75447574245790, −14.98352593483745, −14.73942362854443, −13.89862656599360, −13.01343184351969, −12.67904564650522, −12.09634531151427, −11.43012830812705, −10.88486336487906, −10.23648669113075, −9.869457406214311, −9.170685890219692, −8.538879154057791, −7.756960826574000, −7.275927021374302, −6.620221624331764, −5.951391852475012, −5.283776337112100, −4.636476219593065, −3.570774295569640, −2.914323653552182, −1.943291752599454, −0.9893770211310636, 0,
0.9893770211310636, 1.943291752599454, 2.914323653552182, 3.570774295569640, 4.636476219593065, 5.283776337112100, 5.951391852475012, 6.620221624331764, 7.275927021374302, 7.756960826574000, 8.538879154057791, 9.170685890219692, 9.869457406214311, 10.23648669113075, 10.88486336487906, 11.43012830812705, 12.09634531151427, 12.67904564650522, 13.01343184351969, 13.89862656599360, 14.73942362854443, 14.98352593483745, 15.75447574245790, 16.32684581323117, 16.89185000147877
