| L(s) = 1 | + 3.53·2-s − 3·3-s + 4.46·4-s − 10.5·6-s + 7·7-s − 12.4·8-s + 9·9-s − 2.93·11-s − 13.4·12-s + 19.0·13-s + 24.7·14-s − 79.7·16-s − 122.·17-s + 31.7·18-s + 107.·19-s − 21·21-s − 10.3·22-s − 210.·23-s + 37.4·24-s + 67.3·26-s − 27·27-s + 31.2·28-s + 95.4·29-s − 94.3·31-s − 181.·32-s + 8.81·33-s − 432.·34-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.577·3-s + 0.558·4-s − 0.720·6-s + 0.377·7-s − 0.551·8-s + 0.333·9-s − 0.0805·11-s − 0.322·12-s + 0.406·13-s + 0.471·14-s − 1.24·16-s − 1.74·17-s + 0.416·18-s + 1.29·19-s − 0.218·21-s − 0.100·22-s − 1.90·23-s + 0.318·24-s + 0.507·26-s − 0.192·27-s + 0.211·28-s + 0.611·29-s − 0.546·31-s − 1.00·32-s + 0.0464·33-s − 2.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 3.53T + 8T^{2} \) |
| 11 | \( 1 + 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−10.22830229563661867924912411589, −9.151774616763100073637836451075, −8.130216543569479799823416786368, −6.84290374163458097002565157603, −6.09193953812086260274418914693, −5.15502211710477551068967189687, −4.40949664216371750983075889168, −3.39507048704496452027599025816, −1.92262791641916818510829055902, 0,
1.92262791641916818510829055902, 3.39507048704496452027599025816, 4.40949664216371750983075889168, 5.15502211710477551068967189687, 6.09193953812086260274418914693, 6.84290374163458097002565157603, 8.130216543569479799823416786368, 9.151774616763100073637836451075, 10.22830229563661867924912411589
