| 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
−9.286370759522316377060242557390, −8.540382561994947102004238401816, −7.914039854549605379083817281527, −7.15879862183014807671072531939, −5.63575530967098901895621248826, −4.65386476908546694586810803412, −4.23762079255732129452267719942, −2.65305002834628129910386032758, −1.55870166046766624206783098847, 0,
1.55870166046766624206783098847, 2.65305002834628129910386032758, 4.23762079255732129452267719942, 4.65386476908546694586810803412, 5.63575530967098901895621248826, 7.15879862183014807671072531939, 7.914039854549605379083817281527, 8.540382561994947102004238401816, 9.286370759522316377060242557390
