| L(s) = 1 | − 1.58·2-s + 5.24·3-s − 5.48·4-s − 5·5-s − 8.31·6-s + 21.3·8-s + 0.485·9-s + 7.92·10-s + 28.1·11-s − 28.7·12-s + 3.85·13-s − 26.2·15-s + 9.97·16-s − 38.3·17-s − 0.769·18-s − 116.·19-s + 27.4·20-s − 44.6·22-s − 176.·23-s + 112.·24-s + 25·25-s − 6.11·26-s − 139.·27-s − 209.·29-s + 41.5·30-s + 207.·31-s − 186.·32-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + 1.00·3-s − 0.685·4-s − 0.447·5-s − 0.565·6-s + 0.945·8-s + 0.0179·9-s + 0.250·10-s + 0.771·11-s − 0.691·12-s + 0.0823·13-s − 0.451·15-s + 0.155·16-s − 0.547·17-s − 0.0100·18-s − 1.40·19-s + 0.306·20-s − 0.432·22-s − 1.59·23-s + 0.953·24-s + 0.200·25-s − 0.0461·26-s − 0.990·27-s − 1.34·29-s + 0.252·30-s + 1.20·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.58T + 8T^{2} \) |
| 3 | \( 1 - 5.24T + 27T^{2} \) |
| 11 | \( 1 - 28.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.85T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 15.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 43.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 855.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 252.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 960.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−11.00447198868702718049414150212, −9.906659673113176271002140137805, −9.017159795864714526674431420318, −8.405125639031050811305536291883, −7.65400117964934738215036689460, −6.22068040978213335311116041835, −4.47875616141197994451195542212, −3.63623616393857740721138000635, −1.93756783274652489068805309255, 0,
1.93756783274652489068805309255, 3.63623616393857740721138000635, 4.47875616141197994451195542212, 6.22068040978213335311116041835, 7.65400117964934738215036689460, 8.405125639031050811305536291883, 9.017159795864714526674431420318, 9.906659673113176271002140137805, 11.00447198868702718049414150212
