| L(s) = 1 | + 0.368·2-s − 3·3-s − 7.86·4-s − 1.10·6-s − 26.5·7-s − 5.84·8-s + 9·9-s − 11·11-s + 23.5·12-s + 50.4·13-s − 9.79·14-s + 60.7·16-s + 108.·17-s + 3.31·18-s + 19.1·19-s + 79.7·21-s − 4.05·22-s + 60.4·23-s + 17.5·24-s + 18.5·26-s − 27·27-s + 209.·28-s − 39.2·29-s − 22.4·31-s + 69.1·32-s + 33·33-s + 40.1·34-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 0.577·3-s − 0.983·4-s − 0.0752·6-s − 1.43·7-s − 0.258·8-s + 0.333·9-s − 0.301·11-s + 0.567·12-s + 1.07·13-s − 0.187·14-s + 0.949·16-s + 1.55·17-s + 0.0434·18-s + 0.230·19-s + 0.828·21-s − 0.0392·22-s + 0.547·23-s + 0.149·24-s + 0.140·26-s − 0.192·27-s + 1.41·28-s − 0.251·29-s − 0.130·31-s + 0.382·32-s + 0.174·33-s + 0.202·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.368T + 8T^{2} \) |
| 7 | \( 1 + 26.5T + 343T^{2} \) |
| 13 | \( 1 - 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 60.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 22.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 345.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 96.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 514.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 131.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 68.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 202.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 840.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−9.546720849430344422968443066047, −8.713334417093635033259014746216, −7.70238517580861063544180081923, −6.60557088285084411923632665197, −5.78346847481023724468215391881, −5.10123940286875499913948407614, −3.75635508958991067922495035146, −3.21394321856293129443582605699, −1.11702908847114045941691589691, 0,
1.11702908847114045941691589691, 3.21394321856293129443582605699, 3.75635508958991067922495035146, 5.10123940286875499913948407614, 5.78346847481023724468215391881, 6.60557088285084411923632665197, 7.70238517580861063544180081923, 8.713334417093635033259014746216, 9.546720849430344422968443066047
