| L(s) = 1 | + (−0.754 + 1.30i)2-s + (−0.821 + 0.298i)3-s + (−0.138 − 0.239i)4-s + (1.83 + 1.53i)5-s + (0.228 − 1.29i)6-s + (−0.324 − 0.272i)7-s − 2.60·8-s + (−1.71 + 1.43i)9-s + (−3.39 + 1.23i)10-s + (−2.57 − 0.938i)11-s + (0.184 + 0.155i)12-s + (4.74 + 3.98i)13-s + (0.601 − 0.218i)14-s + (−1.96 − 0.715i)15-s + (2.23 − 3.87i)16-s + (0.603 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.533 + 0.923i)2-s + (−0.474 + 0.172i)3-s + (−0.0690 − 0.119i)4-s + (0.819 + 0.687i)5-s + (0.0934 − 0.530i)6-s + (−0.122 − 0.103i)7-s − 0.919·8-s + (−0.571 + 0.479i)9-s + (−1.07 + 0.390i)10-s + (−0.777 − 0.282i)11-s + (0.0534 + 0.0448i)12-s + (1.31 + 1.10i)13-s + (0.160 − 0.0585i)14-s + (−0.507 − 0.184i)15-s + (0.559 − 0.969i)16-s + (0.146 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.325370 + 0.665955i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325370 + 0.665955i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-9.56 - 4.17i)T \) |
| good | 2 | \( 1 + (0.754 - 1.30i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.821 - 0.298i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-1.83 - 1.53i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.324 + 0.272i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (2.57 + 0.938i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.74 - 3.98i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.603 + 1.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 6.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 0.480i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 1.74i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (4.34 - 3.64i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 + (-2.53 + 4.38i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.393 - 2.23i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.19 + 2.68i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (5.28 + 1.92i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.947 - 5.37i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.27 - 3.58i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.26 + 9.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.2 - 4.44i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.590 + 0.495i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (13.2 - 4.80i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.404 + 2.29i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (14.0 + 11.8i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.96275833801038029400933113278, −13.47709120939965289994150478177, −11.61863152875148860871081898878, −10.94790061370994352044964130451, −9.605042322396477941674439786953, −8.611268235425317693762284939140, −7.27140681696996129009155280121, −6.30545763582777404395242344717, −5.34717229985528918871668564326, −2.93670203848004012695720197560,
1.16628538928944126708868012417, 3.06328989612631805233343701657, 5.48215174198642693215234397366, 6.14799664759867736424175213649, 8.265064143750225432413342806002, 9.223115880322296239041015583777, 10.32190574934548483893410401255, 10.99584249995105243512023582360, 12.37895947497653934891374664494, 12.77580662231711843621620922905
