| L(s) = 1 | + (1.67 + 15.4i)3-s + 6.76i·5-s − 132. i·7-s + (−237. + 51.9i)9-s − 687.·11-s − 609.·13-s + (−104. + 11.3i)15-s + 550. i·17-s + 232. i·19-s + (2.05e3 − 222. i)21-s + 4.48e3·23-s + 3.07e3·25-s + (−1.20e3 − 3.59e3i)27-s − 2.04e3i·29-s + 6.67e3i·31-s + ⋯ |
| L(s) = 1 | + (0.107 + 0.994i)3-s + 0.121i·5-s − 1.02i·7-s + (−0.976 + 0.213i)9-s − 1.71·11-s − 1.00·13-s + (−0.120 + 0.0130i)15-s + 0.462i·17-s + 0.147i·19-s + (1.01 − 0.109i)21-s + 1.76·23-s + 0.985·25-s + (−0.317 − 0.948i)27-s − 0.451i·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.435161275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435161275\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.67 - 15.4i)T \) |
| good | 5 | \( 1 - 6.76iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 132. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 687.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 609.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 550. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 232. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.04e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.67e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 9.78e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.24e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.33e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.81e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.49e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.92e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−10.64844821650341862512258392689, −9.845423611990575789493429580350, −8.764436456757718953580680202064, −7.77017986425181024813829773443, −6.89609937764988904010179525129, −5.29818237229188670752232267727, −4.76958986779796204972643225668, −3.47762613411128602412690835425, −2.51789710922476097041372378768, −0.50831926072943220782353335932,
0.73630190093519450803123198803, 2.41578543044286406865003328087, 2.81901564734455615918486686153, 5.01385753264585417120104856012, 5.55902200001296456901744971789, 6.90920991271760189903361994404, 7.64069310495874434434045290827, 8.585211970925181062408335654939, 9.368828353399343775777406537587, 10.64032848090815186931783758709
