| L(s) = 1 | + (−2.25 + 15.4i)3-s + 86.6i·5-s − 110. i·7-s + (−232. − 69.5i)9-s + 212.·11-s + 539.·13-s + (−1.33e3 − 195. i)15-s + 24.1i·17-s − 1.93e3i·19-s + (1.70e3 + 249. i)21-s − 275.·23-s − 4.38e3·25-s + (1.59e3 − 3.43e3i)27-s − 7.33e3i·29-s − 7.47e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.144 + 0.989i)3-s + 1.55i·5-s − 0.852i·7-s + (−0.958 − 0.286i)9-s + 0.528·11-s + 0.885·13-s + (−1.53 − 0.224i)15-s + 0.0202i·17-s − 1.22i·19-s + (0.843 + 0.123i)21-s − 0.108·23-s − 1.40·25-s + (0.421 − 0.906i)27-s − 1.62i·29-s − 1.39i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.528336035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528336035\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.25 - 15.4i)T \) |
| good | 5 | \( 1 - 86.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 110. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 212.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 539.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 24.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.93e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 275.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.33e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.47e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 7.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.93e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 4.49e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.67e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.55e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.05e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 4.23e3T + 8.58e9T^{2} \) |
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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
−10.59248969771573457777596796113, −9.832023339953009411306415541255, −8.848231476116073018014414924441, −7.55203031311851117941119513586, −6.64059783443468643033526162857, −5.81309775479536112158237832855, −4.24888337005615856413983278035, −3.61120303070461358175998381933, −2.47443375914096221312670601817, −0.43272909945410814649252596021,
1.11732571269374455321423035253, 1.71883072276217681838558157077, 3.41493459680160508860325861394, 4.94574567798718334897733714574, 5.71319315772685926788804996870, 6.63115853792131455816078786155, 8.000060169220384891491437634807, 8.658417187744245736320502652951, 9.194121409893569746728168129731, 10.68422855496964962675372060149
