| L(s) = 1 | + 2i·3-s + 49i·7-s + 239·9-s − 720·11-s − 572i·13-s + 1.25e3i·17-s + 94·19-s − 98·21-s − 96i·23-s + 964i·27-s + 4.37e3·29-s − 6.24e3·31-s − 1.44e3i·33-s − 1.07e4i·37-s + 1.14e3·39-s + ⋯ |
| L(s) = 1 | + 0.128i·3-s + 0.377i·7-s + 0.983·9-s − 1.79·11-s − 0.938i·13-s + 1.05i·17-s + 0.0597·19-s − 0.0484·21-s − 0.0378i·23-s + 0.254i·27-s + 0.965·29-s − 1.16·31-s − 0.230i·33-s − 1.29i·37-s + 0.120·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.751906671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751906671\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
| good | 3 | \( 1 - 2iT - 243T^{2} \) |
| 11 | \( 1 + 720T + 1.61e5T^{2} \) |
| 13 | \( 1 + 572iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 94T + 2.47e6T^{2} \) |
| 23 | \( 1 + 96iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.16e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.58e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.01e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.59e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.11e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.34e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.24e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.52e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.07e5iT - 8.58e9T^{2} \) |
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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
−9.832570293895776412158818852839, −8.649360719332333577174406183918, −7.88655519999573719184279539080, −7.16343352796214164402195215981, −5.85450451788696116084132978317, −5.21057904183596230082472839860, −4.11241372451629510421263495628, −2.95206904774621332742341164589, −1.92846793092497751727320626817, −0.49616688747952715104488793260,
0.75869988972999107093360258247, 2.02340381905979202935592871561, 3.08161184393776156385168972222, 4.42484505528440347936946747270, 5.05436040062549233605444697164, 6.30834623855936092748459620186, 7.32822126539191534170519903721, 7.73690482657648142927380389997, 8.958250196990659294690422451671, 9.848457870421664957455084023860
