| L(s) = 1 | + (−1.62 − 1.17i)2-s + (−1.06 − 3.27i)3-s + (0.626 + 1.92i)4-s + (1.50 + 1.65i)5-s + (−2.13 + 6.57i)6-s + 4.84·7-s + (0.0169 − 0.0523i)8-s + (−7.17 + 5.21i)9-s + (−0.503 − 4.45i)10-s + (2.24 + 1.62i)11-s + (5.65 − 4.10i)12-s + (0.559 − 0.406i)13-s + (−7.87 − 5.71i)14-s + (3.80 − 6.70i)15-s + (3.19 − 2.31i)16-s + (1.43 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.834i)2-s + (−0.614 − 1.89i)3-s + (0.313 + 0.964i)4-s + (0.674 + 0.737i)5-s + (−0.872 + 2.68i)6-s + 1.83·7-s + (0.00601 − 0.0184i)8-s + (−2.39 + 1.73i)9-s + (−0.159 − 1.41i)10-s + (0.675 + 0.491i)11-s + (1.63 − 1.18i)12-s + (0.155 − 0.112i)13-s + (−2.10 − 1.52i)14-s + (0.981 − 1.73i)15-s + (0.797 − 0.579i)16-s + (0.348 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.313609 - 0.823755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313609 - 0.823755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.50 - 1.65i)T \) |
| 23 | \( 1 + (0.809 + 0.587i)T \) |
| good | 2 | \( 1 + (1.62 + 1.17i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (1.06 + 3.27i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.84T + 7T^{2} \) |
| 11 | \( 1 + (-2.24 - 1.62i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.559 + 0.406i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.43 + 4.41i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.270 + 0.831i)T + (-15.3 - 11.1i)T^{2} \) |
| 29 | \( 1 + (0.922 + 2.83i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.44 + 4.43i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.63 + 1.19i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.02 - 0.747i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 + (0.286 + 0.882i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.859 + 2.64i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.08 - 4.42i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.198 - 0.144i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.26 + 13.1i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.31 - 10.1i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.106 + 0.0775i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.141 - 0.436i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.72 - 8.38i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.29 - 0.941i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.431 - 1.32i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.74497303444289595581989330548, −9.547754320903500257819502361238, −8.479548938923499964341009235031, −7.71832135608678503633475022920, −7.17563318329124734302855645332, −6.00268728389572401954074852578, −5.09773824427836636707621825268, −2.55321212046530298671004073472, −1.88480533113834395395054947806, −1.00558029830066045980813061372,
1.25124803301388548219636770436, 3.80757059017833237082425298184, 4.75287755315838264519066380050, 5.58755745555218203940020923304, 6.27486617915635478617950691327, 8.001552405678558872155909783070, 8.717143396806539107070389733448, 9.089337778870404788599621059060, 10.09426904091268059981011425084, 10.66810897608378484630655829608
