| L(s) = 1 | + 0.170·2-s + 3·3-s − 7.97·4-s − 3.72·5-s + 0.512·6-s − 19.4·7-s − 2.72·8-s + 9·9-s − 0.636·10-s − 11·11-s − 23.9·12-s − 72.7·13-s − 3.31·14-s − 11.1·15-s + 63.3·16-s − 30.9·17-s + 1.53·18-s + 35.2·19-s + 29.6·20-s − 58.3·21-s − 1.87·22-s − 17.9·23-s − 8.18·24-s − 111.·25-s − 12.4·26-s + 27·27-s + 154.·28-s + ⋯ |
| L(s) = 1 | + 0.0603·2-s + 0.577·3-s − 0.996·4-s − 0.333·5-s + 0.0348·6-s − 1.04·7-s − 0.120·8-s + 0.333·9-s − 0.0201·10-s − 0.301·11-s − 0.575·12-s − 1.55·13-s − 0.0633·14-s − 0.192·15-s + 0.989·16-s − 0.441·17-s + 0.0201·18-s + 0.425·19-s + 0.331·20-s − 0.605·21-s − 0.0182·22-s − 0.162·23-s − 0.0696·24-s − 0.888·25-s − 0.0937·26-s + 0.192·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3926954899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3926954899\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 0.170T + 8T^{2} \) |
| 5 | \( 1 + 3.72T + 125T^{2} \) |
| 7 | \( 1 + 19.4T + 343T^{2} \) |
| 13 | \( 1 + 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 49.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 43.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 684.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 731.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 38.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 9.94T + 3.57e5T^{2} \) |
| 73 | \( 1 - 591.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 17.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 450.T + 9.12e5T^{2} \) |
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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
−9.067565145417953575072335617023, −7.890495142203804600393203566129, −7.51138947151338439179126357074, −6.49231156342525194871840032233, −5.42461346209337268375494794001, −4.68992758939065396075929819056, −3.73713276902048934730042246651, −3.11438222178984040771786228991, −1.96587793963329195093397288177, −0.26383025929639016343740443772,
0.26383025929639016343740443772, 1.96587793963329195093397288177, 3.11438222178984040771786228991, 3.73713276902048934730042246651, 4.68992758939065396075929819056, 5.42461346209337268375494794001, 6.49231156342525194871840032233, 7.51138947151338439179126357074, 7.890495142203804600393203566129, 9.067565145417953575072335617023
