| L(s) = 1 | + 5.32·2-s − 3·3-s + 20.3·4-s − 19.7·5-s − 15.9·6-s + 25.0·7-s + 65.7·8-s + 9·9-s − 105.·10-s + 11·11-s − 61.0·12-s − 52.0·13-s + 133.·14-s + 59.2·15-s + 187.·16-s − 107.·17-s + 47.9·18-s − 21.1·19-s − 401.·20-s − 75.2·21-s + 58.5·22-s + 48.1·23-s − 197.·24-s + 264.·25-s − 276.·26-s − 27·27-s + 510.·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.54·4-s − 1.76·5-s − 1.08·6-s + 1.35·7-s + 2.90·8-s + 0.333·9-s − 3.32·10-s + 0.301·11-s − 1.46·12-s − 1.10·13-s + 2.55·14-s + 1.01·15-s + 2.92·16-s − 1.52·17-s + 0.627·18-s − 0.254·19-s − 4.49·20-s − 0.782·21-s + 0.567·22-s + 0.436·23-s − 1.67·24-s + 2.11·25-s − 2.08·26-s − 0.192·27-s + 3.44·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.32T + 8T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 - 25.0T + 343T^{2} \) |
| 13 | \( 1 + 52.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 23.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 418.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 10.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 59.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 377.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 471.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 572.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 540.T + 9.12e5T^{2} \) |
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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
−7.940869531177571095107075155152, −7.27719939657057591501303989500, −6.88360518310935421717113162937, −5.68345937996592076083126274577, −4.82299719227640380469682145662, −4.44496442519008939396965475551, −3.87006852613309546909553900789, −2.69710883275081404735826862604, −1.61878668937203328630127954022, 0,
1.61878668937203328630127954022, 2.69710883275081404735826862604, 3.87006852613309546909553900789, 4.44496442519008939396965475551, 4.82299719227640380469682145662, 5.68345937996592076083126274577, 6.88360518310935421717113162937, 7.27719939657057591501303989500, 7.940869531177571095107075155152
