| L(s) = 1 | + 0.130·2-s + 8.99·3-s − 7.98·4-s − 5·5-s + 1.17·6-s + 14.2·7-s − 2.08·8-s + 53.8·9-s − 0.653·10-s − 11·11-s − 71.7·12-s − 57.3·13-s + 1.86·14-s − 44.9·15-s + 63.5·16-s − 75.6·17-s + 7.03·18-s + 19·19-s + 39.9·20-s + 128.·21-s − 1.43·22-s − 79.0·23-s − 18.7·24-s + 25·25-s − 7.49·26-s + 241.·27-s − 113.·28-s + ⋯ |
| L(s) = 1 | + 0.0462·2-s + 1.73·3-s − 0.997·4-s − 0.447·5-s + 0.0799·6-s + 0.769·7-s − 0.0923·8-s + 1.99·9-s − 0.0206·10-s − 0.301·11-s − 1.72·12-s − 1.22·13-s + 0.0355·14-s − 0.773·15-s + 0.993·16-s − 1.07·17-s + 0.0921·18-s + 0.229·19-s + 0.446·20-s + 1.33·21-s − 0.0139·22-s − 0.716·23-s − 0.159·24-s + 0.200·25-s − 0.0565·26-s + 1.72·27-s − 0.767·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 0.130T + 8T^{2} \) |
| 3 | \( 1 - 8.99T + 27T^{2} \) |
| 7 | \( 1 - 14.2T + 343T^{2} \) |
| 13 | \( 1 + 57.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.6T + 4.91e3T^{2} \) |
| 23 | \( 1 + 79.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 218.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 343.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 350.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 65.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 107.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 594.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 130.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 47.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 694.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 898.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 900.T + 9.12e5T^{2} \) |
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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
−9.066535919212173394711684315076, −8.272941275726247314684156219615, −7.81000125525732145483906219727, −7.04149117120496477720099912241, −5.31418190093370568287133745767, −4.45847057452440453274656371508, −3.81150663741515591874797485024, −2.71588910270787587134679155833, −1.72871328222877344229043683398, 0,
1.72871328222877344229043683398, 2.71588910270787587134679155833, 3.81150663741515591874797485024, 4.45847057452440453274656371508, 5.31418190093370568287133745767, 7.04149117120496477720099912241, 7.81000125525732145483906219727, 8.272941275726247314684156219615, 9.066535919212173394711684315076
