| L(s) = 1 | + (0.877 − 0.479i)2-s + (−0.997 + 0.0713i)3-s + (0.540 − 0.841i)4-s + (−1.84 + 1.26i)5-s + (−0.841 + 0.540i)6-s + (1.33 − 3.57i)7-s + (0.0713 − 0.997i)8-s + (0.989 − 0.142i)9-s + (−1.01 + 1.99i)10-s + (−0.290 − 0.987i)11-s + (−0.479 + 0.877i)12-s + (−3.47 + 1.29i)13-s + (−0.542 − 3.77i)14-s + (1.75 − 1.39i)15-s + (−0.415 − 0.909i)16-s + (4.48 + 0.975i)17-s + ⋯ |
| L(s) = 1 | + (0.620 − 0.338i)2-s + (−0.575 + 0.0411i)3-s + (0.270 − 0.420i)4-s + (−0.825 + 0.564i)5-s + (−0.343 + 0.220i)6-s + (0.503 − 1.35i)7-s + (0.0252 − 0.352i)8-s + (0.329 − 0.0474i)9-s + (−0.320 + 0.630i)10-s + (−0.0874 − 0.297i)11-s + (−0.138 + 0.253i)12-s + (−0.964 + 0.359i)13-s + (−0.145 − 1.00i)14-s + (0.451 − 0.359i)15-s + (−0.103 − 0.227i)16-s + (1.08 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.422259 - 0.987941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422259 - 0.987941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.877 + 0.479i)T \) |
| 3 | \( 1 + (0.997 - 0.0713i)T \) |
| 5 | \( 1 + (1.84 - 1.26i)T \) |
| 23 | \( 1 + (3.71 + 3.03i)T \) |
| good | 7 | \( 1 + (-1.33 + 3.57i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (0.290 + 0.987i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (3.47 - 1.29i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (-4.48 - 0.975i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (6.05 + 3.89i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.296 + 0.461i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (7.16 + 8.27i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-2.74 + 3.66i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.389 + 2.70i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.101 - 1.41i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (-3.43 + 3.43i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.86 - 1.06i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (-1.04 - 0.479i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.855 - 0.741i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 6.65i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (-4.57 - 1.34i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.30 - 10.5i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (0.501 - 1.09i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-9.90 - 7.41i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (3.18 - 3.67i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (13.7 - 10.2i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.61328574585386365606557415914, −9.662415862832277634419253514492, −8.121054246455591828173182525156, −7.33847158556151510365858949124, −6.67052324533997473507155795325, −5.48725455336472528392562485151, −4.25211352918244152391582994001, −3.96191802249865218784401800227, −2.35280572772723715941998858847, −0.48258309200147988780151633211,
1.92656003490788920349970832071, 3.43123494655281303286695965706, 4.67292216534126954304178413084, 5.29953241331979230447407418214, 6.05589200679983630224394462742, 7.36882615756081699597691655403, 8.016723534730634929892116535676, 8.853073101946811827415559911872, 9.980854869638364045262344433534, 11.07357218071789115054913665078
