| L(s) = 1 | + 9.91·2-s − 9·3-s + 66.3·4-s − 25·5-s − 89.2·6-s + 92.6·7-s + 340.·8-s + 81·9-s − 247.·10-s + 121·11-s − 597.·12-s + 800.·13-s + 918.·14-s + 225·15-s + 1.25e3·16-s − 117.·17-s + 803.·18-s + 831.·19-s − 1.65e3·20-s − 833.·21-s + 1.20e3·22-s + 2.95e3·23-s − 3.06e3·24-s + 625·25-s + 7.93e3·26-s − 729·27-s + 6.14e3·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.07·4-s − 0.447·5-s − 1.01·6-s + 0.714·7-s + 1.88·8-s + 0.333·9-s − 0.784·10-s + 0.301·11-s − 1.19·12-s + 1.31·13-s + 1.25·14-s + 0.258·15-s + 1.22·16-s − 0.0988·17-s + 0.584·18-s + 0.528·19-s − 0.927·20-s − 0.412·21-s + 0.528·22-s + 1.16·23-s − 1.08·24-s + 0.200·25-s + 2.30·26-s − 0.192·27-s + 1.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.093431563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.093431563\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 9.91T + 32T^{2} \) |
| 7 | \( 1 - 92.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 800.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 117.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 831.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 61.7T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.59e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.74e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.98e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.48e4T + 8.58e9T^{2} \) |
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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
−11.94872078451758627463333353908, −11.36700150862993840866874312589, −10.58893377518855998047145526780, −8.656401738345292073450692192141, −7.23804348541416272608671683331, −6.26807381796681273112661186578, −5.20160616164181060195187458865, −4.31178835262205792821061783099, −3.17794485127400430122486985968, −1.34916322237558547301002581774,
1.34916322237558547301002581774, 3.17794485127400430122486985968, 4.31178835262205792821061783099, 5.20160616164181060195187458865, 6.26807381796681273112661186578, 7.23804348541416272608671683331, 8.656401738345292073450692192141, 10.58893377518855998047145526780, 11.36700150862993840866874312589, 11.94872078451758627463333353908
