| L(s) = 1 | + (−10.7 + 11.3i)3-s + 4.25·5-s + 187. i·7-s + (−12.5 − 242. i)9-s − 712. i·11-s + 804. i·13-s + (−45.6 + 48.0i)15-s + 309. i·17-s + 2.46e3·19-s + (−2.11e3 − 2.01e3i)21-s + 3.06e3·23-s − 3.10e3·25-s + (2.87e3 + 2.46e3i)27-s + 6.32e3·29-s − 9.00e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.688 + 0.725i)3-s + 0.0760·5-s + 1.44i·7-s + (−0.0516 − 0.998i)9-s − 1.77i·11-s + 1.31i·13-s + (−0.0523 + 0.0551i)15-s + 0.259i·17-s + 1.56·19-s + (−1.04 − 0.995i)21-s + 1.20·23-s − 0.994·25-s + (0.759 + 0.650i)27-s + 1.39·29-s − 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0258 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0258 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.614944618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614944618\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (10.7 - 11.3i)T \) |
| good | 5 | \( 1 - 4.25T + 3.12e3T^{2} \) |
| 7 | \( 1 - 187. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 712. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 804. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 309. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.00e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.35e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.10e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.07e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 9.40e4T + 8.58e9T^{2} \) |
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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
−10.98320651681920400315518121439, −9.660411116283327722063962397459, −9.107665244415281897378018163703, −8.255329563788211014789863687478, −6.61155495335825710223822766865, −5.81134161825079970764704462249, −5.14985364269647583193358288459, −3.78128567838486896563249048533, −2.68097876839876805829096707366, −0.928015746372507847796980616538,
0.60068019769877229606578955246, 1.47424195506936382871063760277, 3.06512306152046434260294717675, 4.60732951069689921941201024690, 5.32860792174506152126348654035, 6.80714783961320520694674576130, 7.29484035844656272400365522837, 8.020377160502182265568285800649, 9.734762766350996738693750541004, 10.30739771346935635678398923210
