| L(s) = 1 | + (4.08 − 8.01i)3-s + 47.0i·5-s − 13.4·7-s + (−47.6 − 65.5i)9-s + 135. i·11-s + 80.8·13-s + (377. + 192. i)15-s − 430. i·17-s − 43.5·19-s + (−54.9 + 107. i)21-s + 873. i·23-s − 1.59e3·25-s + (−719. + 114. i)27-s − 707. i·29-s − 677.·31-s + ⋯ |
| L(s) = 1 | + (0.453 − 0.891i)3-s + 1.88i·5-s − 0.274·7-s + (−0.587 − 0.808i)9-s + 1.11i·11-s + 0.478·13-s + (1.67 + 0.854i)15-s − 1.49i·17-s − 0.120·19-s + (−0.124 + 0.244i)21-s + 1.65i·23-s − 2.54·25-s + (−0.987 + 0.156i)27-s − 0.840i·29-s − 0.705·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7487119956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7487119956\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.08 + 8.01i)T \) |
| good | 5 | \( 1 - 47.0iT - 625T^{2} \) |
| 7 | \( 1 + 13.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 135. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 80.8T + 2.85e4T^{2} \) |
| 17 | \( 1 + 430. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 43.5T + 1.30e5T^{2} \) |
| 23 | \( 1 - 873. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 707. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 677.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.22e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.43e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.06e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.46e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 470. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 39.9T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.52e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.85e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.27e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.93e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.27e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.78e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
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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
−11.37235616806347582424663387367, −10.04791431896270533555921861183, −9.471441340989325672062911472624, −8.010528678108602238247818637179, −7.01348055057080141972743918158, −6.90423442754042688975360032176, −5.60096877177846163810711174116, −3.68348707771573962509108718942, −2.83259918029026045540194409898, −1.82741308516799639282854981663,
0.18669832733566958953144983474, 1.64172265217347465589443968833, 3.42405693536155823518794653129, 4.31956098585692664252997192867, 5.27181589225700384668260270208, 6.15860510658584542853028141756, 8.104776198630411913683048382677, 8.644599389341565103452860097016, 9.087548296436288183018546117941, 10.29152885924489359404821697281
