| L(s) = 1 | + (28.3 + 28.3i)5-s − 55.5i·7-s + (−137. − 137. i)11-s + (−574. + 574. i)13-s − 320.·17-s + (858. − 858. i)19-s + 825. i·23-s − 1.52e3i·25-s + (−333. + 333. i)29-s + 8.90e3·31-s + (1.57e3 − 1.57e3i)35-s + (−3.87e3 − 3.87e3i)37-s − 6.54e3i·41-s + (5.62e3 + 5.62e3i)43-s − 3.12e3·47-s + ⋯ |
| L(s) = 1 | + (0.506 + 0.506i)5-s − 0.428i·7-s + (−0.342 − 0.342i)11-s + (−0.942 + 0.942i)13-s − 0.269·17-s + (0.545 − 0.545i)19-s + 0.325i·23-s − 0.486i·25-s + (−0.0737 + 0.0737i)29-s + 1.66·31-s + (0.217 − 0.217i)35-s + (−0.465 − 0.465i)37-s − 0.608i·41-s + (0.463 + 0.463i)43-s − 0.206·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.000621417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.000621417\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-28.3 - 28.3i)T + 3.12e3iT^{2} \) |
| 7 | \( 1 + 55.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (137. + 137. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (574. - 574. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + 320.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-858. + 858. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 825. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (333. - 333. i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 - 8.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (3.87e3 + 3.87e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 6.54e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-5.62e3 - 5.62e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 3.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.81e4 + 1.81e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (5.53e3 + 5.53e3i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (701. - 701. i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (-1.44e4 + 1.44e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.36e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.77e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.89e4 + 7.89e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 2.59e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 3.91e4T + 8.58e9T^{2} \) |
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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
−9.894163130880140071694170610430, −9.159425612360609567745189617879, −8.022502572916957748167823558982, −7.06580064844127224712166405063, −6.39833207593914026182206391160, −5.22325551304169993909802243233, −4.26894327872751708584500773722, −2.95647065641830565673284353583, −2.03210820963072237051608051263, −0.55468876648369454234566894238,
0.847080053442613089239918016545, 2.13521533204229090269090349761, 3.12191475228565132853615718811, 4.65576120935773414354019098478, 5.34012123662041944797206225914, 6.25967432354307179692277788953, 7.47345949223774822543612939723, 8.231383739557459234891889386193, 9.255409109210336699930200587992, 9.940555236275510880641785162540
