| 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
−9.940555236275510880641785162540, −9.255409109210336699930200587992, −8.231383739557459234891889386193, −7.47345949223774822543612939723, −6.25967432354307179692277788953, −5.34012123662041944797206225914, −4.65576120935773414354019098478, −3.12191475228565132853615718811, −2.13521533204229090269090349761, −0.847080053442613089239918016545,
0.55468876648369454234566894238, 2.03210820963072237051608051263, 2.95647065641830565673284353583, 4.26894327872751708584500773722, 5.22325551304169993909802243233, 6.39833207593914026182206391160, 7.06580064844127224712166405063, 8.022502572916957748167823558982, 9.159425612360609567745189617879, 9.894163130880140071694170610430
