| L(s) = 1 | − 7.52·2-s − 9·3-s + 24.6·4-s + 67.7·6-s + 234.·7-s + 55.3·8-s + 81·9-s + 121·11-s − 221.·12-s − 236.·13-s − 1.76e3·14-s − 1.20e3·16-s + 608.·17-s − 609.·18-s − 1.79e3·19-s − 2.10e3·21-s − 910.·22-s + 4.77e3·23-s − 498.·24-s + 1.77e3·26-s − 729·27-s + 5.76e3·28-s + 2.80e3·29-s + 1.02e4·31-s + 7.29e3·32-s − 1.08e3·33-s − 4.57e3·34-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 0.577·3-s + 0.770·4-s + 0.768·6-s + 1.80·7-s + 0.305·8-s + 0.333·9-s + 0.301·11-s − 0.444·12-s − 0.387·13-s − 2.40·14-s − 1.17·16-s + 0.510·17-s − 0.443·18-s − 1.14·19-s − 1.04·21-s − 0.401·22-s + 1.88·23-s − 0.176·24-s + 0.515·26-s − 0.192·27-s + 1.39·28-s + 0.619·29-s + 1.91·31-s + 1.26·32-s − 0.174·33-s − 0.678·34-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
| \(L(3)\) |
\(\approx\) |
\(1.305112273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305112273\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 7.52T + 32T^{2} \) |
| 7 | \( 1 - 234.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 236.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 608.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.45e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.34e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.83e5T + 8.58e9T^{2} \) |
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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
−9.393350503129311585913466787420, −8.585891465726003838720228764342, −7.936484749471752512268444197925, −7.23384315806196064997757086848, −6.19631191422080101713873885554, −4.83229650989614724147261327144, −4.49775773130483768514682363301, −2.49164630023142279570935660853, −1.34969873187599283937220738493, −0.76750934555467800063821478776,
0.76750934555467800063821478776, 1.34969873187599283937220738493, 2.49164630023142279570935660853, 4.49775773130483768514682363301, 4.83229650989614724147261327144, 6.19631191422080101713873885554, 7.23384315806196064997757086848, 7.936484749471752512268444197925, 8.585891465726003838720228764342, 9.393350503129311585913466787420
