| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (1.68 − 3.68i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−0.746 − 0.219i)11-s + (−0.959 − 0.281i)12-s + (−1.73 − 3.80i)13-s + (−0.576 + 4.00i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (0.246 + 0.158i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (0.343 + 0.220i)6-s + (0.635 − 1.39i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (−0.224 − 0.0660i)11-s + (−0.276 − 0.0813i)12-s + (−0.482 − 1.05i)13-s + (−0.154 + 1.07i)14-s + (−0.169 + 0.195i)15-s + (0.103 − 0.227i)16-s + (0.0598 + 0.0384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.252580 - 0.675228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.252580 - 0.675228i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-2.10 - 4.31i)T \) |
| good | 7 | \( 1 + (-1.68 + 3.68i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.746 + 0.219i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.73 + 3.80i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.246 - 0.158i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.50 + 0.966i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (2.37 + 1.52i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (5.90 - 6.81i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.01 + 7.03i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.298 - 2.07i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.60 + 5.31i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 + (-1.69 + 3.72i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (3.75 + 8.23i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.280 - 0.324i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.59 - 1.05i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (9.31 - 2.73i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 8.48i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.41 + 3.08i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.601 - 4.18i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (4.76 + 5.49i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.102 - 0.711i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.25812794859685996562797354139, −9.274567160048931627853512893933, −8.156830269800801432769868945503, −7.53674966307573974147242714563, −6.98010848163569500955847934745, −5.59527904127481614969732611162, −4.88189086425228697371486344042, −3.44749835359692879807985397035, −1.67022297383775141978312692175, −0.49847979307353409753437218136,
1.89141276993107501093332249464, 2.93445496291974731473902849182, 4.42083208708231024902748345212, 5.44676980983573743489380162126, 6.38847353313395076204894269614, 7.41024160154467454471396711941, 8.407723052124611253887360210150, 9.187862478004835845188732141516, 9.836578758688484544699771304661, 10.86206348828360293704002436470
