| L(s) = 1 | + 5.01·2-s − 7.14·3-s − 6.85·4-s + 25·5-s − 35.8·6-s − 38.8·7-s − 194.·8-s − 191.·9-s + 125.·10-s − 121·11-s + 48.9·12-s − 458.·13-s − 194.·14-s − 178.·15-s − 757.·16-s + 2.12e3·17-s − 962.·18-s − 361·19-s − 171.·20-s + 277.·21-s − 606.·22-s − 4.77e3·23-s + 1.39e3·24-s + 625·25-s − 2.29e3·26-s + 3.10e3·27-s + 266.·28-s + ⋯ |
| L(s) = 1 | + 0.886·2-s − 0.458·3-s − 0.214·4-s + 0.447·5-s − 0.406·6-s − 0.299·7-s − 1.07·8-s − 0.789·9-s + 0.396·10-s − 0.301·11-s + 0.0981·12-s − 0.752·13-s − 0.265·14-s − 0.205·15-s − 0.739·16-s + 1.78·17-s − 0.700·18-s − 0.229·19-s − 0.0957·20-s + 0.137·21-s − 0.267·22-s − 1.88·23-s + 0.493·24-s + 0.200·25-s − 0.666·26-s + 0.820·27-s + 0.0641·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.013973637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013973637\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 5.01T + 32T^{2} \) |
| 3 | \( 1 + 7.14T + 243T^{2} \) |
| 7 | \( 1 + 38.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 458.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.46e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.75e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.73e3T + 8.58e9T^{2} \) |
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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
−9.435653758503458343814949330726, −8.313123193795644875738658967970, −7.45291456034191113501459931473, −6.16756401204968488630116564535, −5.64306639041247511669808726105, −5.13433582904861471283867980000, −3.90919339372741017093066275164, −3.13057020401590131699528925889, −2.01653407749750773105053208945, −0.36853844690110670900826192042,
0.36853844690110670900826192042, 2.01653407749750773105053208945, 3.13057020401590131699528925889, 3.90919339372741017093066275164, 5.13433582904861471283867980000, 5.64306639041247511669808726105, 6.16756401204968488630116564535, 7.45291456034191113501459931473, 8.313123193795644875738658967970, 9.435653758503458343814949330726
