| L(s) = 1 | + (−4.5 − 7.79i)3-s + 78.9·5-s + (60.7 − 105. i)7-s + (−40.5 + 70.1i)9-s + (−94.0 − 162. i)11-s + (−85.7 − 603. i)13-s + (−355. − 615. i)15-s + (34.3 − 59.4i)17-s + (−33.2 + 57.5i)19-s − 1.09e3·21-s + (−1.71e3 − 2.97e3i)23-s + 3.11e3·25-s + 729·27-s + (103. + 178. i)29-s − 298.·31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + 1.41·5-s + (0.468 − 0.811i)7-s + (−0.166 + 0.288i)9-s + (−0.234 − 0.406i)11-s + (−0.140 − 0.990i)13-s + (−0.407 − 0.706i)15-s + (0.0287 − 0.0498i)17-s + (−0.0211 + 0.0366i)19-s − 0.541·21-s + (−0.676 − 1.17i)23-s + 0.995·25-s + 0.192·27-s + (0.0227 + 0.0394i)29-s − 0.0557·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.085436602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085436602\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 13 | \( 1 + (85.7 + 603. i)T \) |
| good | 5 | \( 1 - 78.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-60.7 + 105. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (94.0 + 162. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 17 | \( 1 + (-34.3 + 59.4i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (33.2 - 57.5i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.71e3 + 2.97e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-103. - 178. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 298.T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-3.82e3 - 6.62e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-3.83e3 - 6.64e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.39e3 + 2.41e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 - 2.21e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.79e4 + 3.10e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.49e3 + 4.31e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.31e4 + 2.27e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.45e4 + 4.25e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + 3.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.04e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (6.96e4 + 1.20e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-3.17e4 + 5.49e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−10.44407140699766963524410036774, −9.836274698387740493940297543443, −8.496370158604677917895739744861, −7.61321187668545313407750526293, −6.42935773900859565428189319975, −5.69471592448529632126531178011, −4.62540281158029818454065429633, −2.87383152276269829663487302513, −1.67398452062337375114753186240, −0.54329635736478658980977238535,
1.60446161747758800850611060852, 2.49395379766538467072298593712, 4.22997985124056789294924808585, 5.41498745920613095419896578435, 5.93221840307397563199020015010, 7.18863226102850714567896906273, 8.634477095942360805889838327669, 9.472576197728451755620411823167, 10.01858612248309490594966074979, 11.15449436474031919288219006288
