| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 26·7-s + 8·8-s + 9·9-s − 10·10-s − 48·11-s + 12·12-s − 88·13-s + 52·14-s − 15·15-s + 16·16-s − 54·17-s + 18·18-s − 160·19-s − 20·20-s + 78·21-s − 96·22-s − 48·23-s + 24·24-s + 25·25-s − 176·26-s + 27·27-s + 104·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.40·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.31·11-s + 0.288·12-s − 1.87·13-s + 0.992·14-s − 0.258·15-s + 1/4·16-s − 0.770·17-s + 0.235·18-s − 1.93·19-s − 0.223·20-s + 0.810·21-s − 0.930·22-s − 0.435·23-s + 0.204·24-s + 1/5·25-s − 1.32·26-s + 0.192·27-s + 0.701·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 31 | \( 1 - p T \) |
| good | 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 88 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 160 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 37 | \( 1 + 304 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 - 30 T + p^{3} T^{2} \) |
| 61 | \( 1 + 478 T + p^{3} T^{2} \) |
| 67 | \( 1 + 334 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1062 T + p^{3} T^{2} \) |
| 73 | \( 1 - 212 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 972 T + p^{3} T^{2} \) |
| 89 | \( 1 - 120 T + p^{3} T^{2} \) |
| 97 | \( 1 - 86 T + p^{3} T^{2} \) |
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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
−9.119609791196702552558972593859, −8.144241714271538468374827859541, −7.68360432394601083667149064837, −6.86770734752692927442033610688, −5.43312311932102694928202949088, −4.71167036938340332771645976958, −4.07788388684637798625410832278, −2.49659195379817187233620172379, −2.08408530339461023258800385508, 0,
2.08408530339461023258800385508, 2.49659195379817187233620172379, 4.07788388684637798625410832278, 4.71167036938340332771645976958, 5.43312311932102694928202949088, 6.86770734752692927442033610688, 7.68360432394601083667149064837, 8.144241714271538468374827859541, 9.119609791196702552558972593859
