| L(s) = 1 | + 1.34·2-s − 19.6·3-s − 30.1·4-s − 25·5-s − 26.4·6-s + 48.6·7-s − 83.6·8-s + 142.·9-s − 33.6·10-s + 283.·11-s + 593.·12-s + 169·13-s + 65.4·14-s + 491.·15-s + 853.·16-s + 789.·17-s + 192.·18-s + 83.3·19-s + 754.·20-s − 955.·21-s + 381.·22-s − 1.93e3·23-s + 1.64e3·24-s + 625·25-s + 227.·26-s + 1.96e3·27-s − 1.46e3·28-s + ⋯ |
| L(s) = 1 | + 0.237·2-s − 1.26·3-s − 0.943·4-s − 0.447·5-s − 0.299·6-s + 0.375·7-s − 0.462·8-s + 0.587·9-s − 0.106·10-s + 0.706·11-s + 1.18·12-s + 0.277·13-s + 0.0892·14-s + 0.563·15-s + 0.833·16-s + 0.662·17-s + 0.139·18-s + 0.0529·19-s + 0.421·20-s − 0.472·21-s + 0.167·22-s − 0.762·23-s + 0.582·24-s + 0.200·25-s + 0.0659·26-s + 0.519·27-s − 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8381062709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8381062709\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 1.34T + 32T^{2} \) |
| 3 | \( 1 + 19.6T + 243T^{2} \) |
| 7 | \( 1 - 48.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 283.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 789.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 83.3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 222.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.36e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.06e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.26e5T + 8.58e9T^{2} \) |
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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
−13.89607476738491918548897703077, −12.51574985788428362753981212146, −11.79983139663780407242990788198, −10.71203451479015943186668990743, −9.332613117427362194318853287454, −7.950740259562945839400802690594, −6.22712299221087313494738312615, −5.12159272704759478332638849424, −3.90113677682194073475758266731, −0.76699392355513239405568926471,
0.76699392355513239405568926471, 3.90113677682194073475758266731, 5.12159272704759478332638849424, 6.22712299221087313494738312615, 7.950740259562945839400802690594, 9.332613117427362194318853287454, 10.71203451479015943186668990743, 11.79983139663780407242990788198, 12.51574985788428362753981212146, 13.89607476738491918548897703077
