| L(s) = 1 | + 2.33·2-s + 9·3-s − 26.5·4-s + 20.9·6-s − 146.·7-s − 136.·8-s + 81·9-s + 121·11-s − 239.·12-s + 158.·13-s − 341.·14-s + 531.·16-s + 1.92e3·17-s + 188.·18-s − 32.9·19-s − 1.31e3·21-s + 282.·22-s − 2.07e3·23-s − 1.22e3·24-s + 369.·26-s + 729·27-s + 3.88e3·28-s + 1.64e3·29-s − 3.56e3·31-s + 5.61e3·32-s + 1.08e3·33-s + 4.49e3·34-s + ⋯ |
| L(s) = 1 | + 0.412·2-s + 0.577·3-s − 0.829·4-s + 0.238·6-s − 1.12·7-s − 0.754·8-s + 0.333·9-s + 0.301·11-s − 0.479·12-s + 0.259·13-s − 0.465·14-s + 0.518·16-s + 1.61·17-s + 0.137·18-s − 0.0209·19-s − 0.651·21-s + 0.124·22-s − 0.818·23-s − 0.435·24-s + 0.107·26-s + 0.192·27-s + 0.936·28-s + 0.362·29-s − 0.666·31-s + 0.968·32-s + 0.174·33-s + 0.666·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 2.33T + 32T^{2} \) |
| 7 | \( 1 + 146.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 158.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 32.9T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.00e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.79e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.65e4T + 8.58e9T^{2} \) |
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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
−9.188946236445906319574153345098, −8.281068766249702067588085710528, −7.42998234661260713286057856536, −6.23486758531879141882411999205, −5.57593871869554587041963759520, −4.31484913506695765515812008384, −3.55307316552815280205947802213, −2.84914741973679579029776514844, −1.21691700057161989784813509351, 0,
1.21691700057161989784813509351, 2.84914741973679579029776514844, 3.55307316552815280205947802213, 4.31484913506695765515812008384, 5.57593871869554587041963759520, 6.23486758531879141882411999205, 7.42998234661260713286057856536, 8.281068766249702067588085710528, 9.188946236445906319574153345098
