| L(s) = 1 | + 0.666·2-s − 10.2·3-s − 7.55·4-s − 5·5-s − 6.82·6-s + 4.34·7-s − 10.3·8-s + 77.9·9-s − 3.33·10-s − 11·11-s + 77.4·12-s + 15.3·13-s + 2.89·14-s + 51.2·15-s + 53.5·16-s − 111.·17-s + 51.8·18-s − 19·19-s + 37.7·20-s − 44.5·21-s − 7.32·22-s − 182.·23-s + 106.·24-s + 25·25-s + 10.2·26-s − 521.·27-s − 32.8·28-s + ⋯ |
| L(s) = 1 | + 0.235·2-s − 1.97·3-s − 0.944·4-s − 0.447·5-s − 0.464·6-s + 0.234·7-s − 0.457·8-s + 2.88·9-s − 0.105·10-s − 0.301·11-s + 1.86·12-s + 0.328·13-s + 0.0552·14-s + 0.881·15-s + 0.836·16-s − 1.58·17-s + 0.679·18-s − 0.229·19-s + 0.422·20-s − 0.462·21-s − 0.0709·22-s − 1.65·23-s + 0.902·24-s + 0.200·25-s + 0.0772·26-s − 3.71·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1148865664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1148865664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 0.666T + 8T^{2} \) |
| 3 | \( 1 + 10.2T + 27T^{2} \) |
| 7 | \( 1 - 4.34T + 343T^{2} \) |
| 13 | \( 1 - 15.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.29T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 42.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 40.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 337.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 694.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 899.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 745.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 866.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 828.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 655.T + 9.12e5T^{2} \) |
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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
−9.812681071403493990255850078669, −8.724178071010148587003120293547, −7.77989881347047793224617958856, −6.68327781553431393409593157363, −6.04233486545570955900228373785, −5.10710393037819070401338800046, −4.53136620567011941016500188155, −3.78898208262795229875854290339, −1.64908234850669410563131189945, −0.19495826052610216765386450106,
0.19495826052610216765386450106, 1.64908234850669410563131189945, 3.78898208262795229875854290339, 4.53136620567011941016500188155, 5.10710393037819070401338800046, 6.04233486545570955900228373785, 6.68327781553431393409593157363, 7.77989881347047793224617958856, 8.724178071010148587003120293547, 9.812681071403493990255850078669
