| L(s) = 1 | + 5.88·2-s − 9·3-s + 2.57·4-s − 52.9·6-s + 219.·7-s − 173.·8-s + 81·9-s + 121·11-s − 23.1·12-s + 842.·13-s + 1.28e3·14-s − 1.09e3·16-s − 1.32e3·17-s + 476.·18-s + 2.03e3·19-s − 1.97e3·21-s + 711.·22-s + 2.82e3·23-s + 1.55e3·24-s + 4.95e3·26-s − 729·27-s + 564.·28-s − 6.72e3·29-s + 2.58e3·31-s − 930.·32-s − 1.08e3·33-s − 7.77e3·34-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0805·4-s − 0.600·6-s + 1.69·7-s − 0.955·8-s + 0.333·9-s + 0.301·11-s − 0.0465·12-s + 1.38·13-s + 1.75·14-s − 1.07·16-s − 1.11·17-s + 0.346·18-s + 1.29·19-s − 0.976·21-s + 0.313·22-s + 1.11·23-s + 0.551·24-s + 1.43·26-s − 0.192·27-s + 0.136·28-s − 1.48·29-s + 0.482·31-s − 0.160·32-s − 0.174·33-s − 1.15·34-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
| \(L(3)\) |
\(\approx\) |
\(3.845380662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.845380662\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 5.88T + 32T^{2} \) |
| 7 | \( 1 - 219.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 842.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.94e4T + 8.58e9T^{2} \) |
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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
−9.309130316850555653685802761985, −8.683956218951869318042464014777, −7.66859159120690633358830723196, −6.60567182744372996633457809038, −5.66289987964876305137500583161, −5.01082408556621755682722266690, −4.30075531480005056897892506344, −3.36439895116074411609613628198, −1.84618915005560917780114568201, −0.829391878551609159693798676385,
0.829391878551609159693798676385, 1.84618915005560917780114568201, 3.36439895116074411609613628198, 4.30075531480005056897892506344, 5.01082408556621755682722266690, 5.66289987964876305137500583161, 6.60567182744372996633457809038, 7.66859159120690633358830723196, 8.683956218951869318042464014777, 9.309130316850555653685802761985
