| L(s) = 1 | + (0.875 + 2.69i)3-s + (82.1 + 59.6i)5-s + (3.44 − 10.5i)7-s + (190. − 138. i)9-s + (−361. + 175. i)11-s + (−218. + 158. i)13-s + (−88.9 + 273. i)15-s + (816. + 593. i)17-s + (186. + 574. i)19-s + 31.5·21-s + 3.24e3·23-s + (2.22e3 + 6.83e3i)25-s + (1.09e3 + 796. i)27-s + (−263. + 810. i)29-s + (−3.97e3 + 2.88e3i)31-s + ⋯ |
| L(s) = 1 | + (0.0561 + 0.172i)3-s + (1.46 + 1.06i)5-s + (0.0265 − 0.0816i)7-s + (0.782 − 0.568i)9-s + (−0.899 + 0.436i)11-s + (−0.358 + 0.260i)13-s + (−0.102 + 0.314i)15-s + (0.685 + 0.497i)17-s + (0.118 + 0.364i)19-s + 0.0156·21-s + 1.27·23-s + (0.710 + 2.18i)25-s + (0.289 + 0.210i)27-s + (−0.0581 + 0.179i)29-s + (−0.741 + 0.539i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.03149 + 1.18453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03149 + 1.18453i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (361. - 175. i)T \) |
| good | 3 | \( 1 + (-0.875 - 2.69i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 + (-82.1 - 59.6i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-3.44 + 10.5i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (218. - 158. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-816. - 593. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-186. - 574. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 3.24e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (263. - 810. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.97e3 - 2.88e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.22e3 + 6.84e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.70e3 - 1.14e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (428. + 1.32e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.15e4 + 8.42e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.56e4 + 4.82e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.77e4 + 2.74e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.37e4 + 1.00e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.71e3 + 1.45e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-7.47e4 + 5.43e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.33e4 + 9.68e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 2.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.06e5 + 7.75e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−13.38773634497734031586053546718, −12.54878984196383238722518988504, −10.84186742257351774620529496339, −10.09340326677248619299698814894, −9.331030421399893201124105660045, −7.43018863239513808854399317956, −6.44135434460409104114846354615, −5.15057429594689107823821395548, −3.19777757807928076882710885657, −1.74891690678107420590820251258,
1.05500368219687923703194439498, 2.48174119719048741759183509265, 4.89738609012613945949906669727, 5.64127594230508005602960600243, 7.31422177843943336498593884358, 8.656917566996273229187398591292, 9.713958973179212221110696483179, 10.57572773507461946311818865440, 12.25550805510626994502666751705, 13.36281525678560216338051512070
