| L(s) = 1 | + (8.99 − 2.40i)2-s + (6.51 − 24.3i)3-s + (47.3 − 27.3i)4-s + (−2.18 + 55.8i)5-s − 234. i·6-s + (−13.7 + 128. i)7-s + (148. − 148. i)8-s + (−338. − 195. i)9-s + (114. + 507. i)10-s + (−255. − 442. i)11-s + (−356. − 1.32e3i)12-s + (632. + 632. i)13-s + (186. + 1.19e3i)14-s + (1.34e3 + 417. i)15-s + (106. − 184. i)16-s + (115. + 30.9i)17-s + ⋯ |
| L(s) = 1 | + (1.58 − 0.425i)2-s + (0.418 − 1.56i)3-s + (1.47 − 0.853i)4-s + (−0.0390 + 0.999i)5-s − 2.65i·6-s + (−0.106 + 0.994i)7-s + (0.822 − 0.822i)8-s + (−1.39 − 0.804i)9-s + (0.363 + 1.60i)10-s + (−0.636 − 1.10i)11-s + (−0.713 − 2.66i)12-s + (1.03 + 1.03i)13-s + (0.254 + 1.62i)14-s + (1.54 + 0.478i)15-s + (0.103 − 0.179i)16-s + (0.0969 + 0.0259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.99424 - 2.25167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.99424 - 2.25167i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.18 - 55.8i)T \) |
| 7 | \( 1 + (13.7 - 128. i)T \) |
| good | 2 | \( 1 + (-8.99 + 2.40i)T + (27.7 - 16i)T^{2} \) |
| 3 | \( 1 + (-6.51 + 24.3i)T + (-210. - 121.5i)T^{2} \) |
| 11 | \( 1 + (255. + 442. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-632. - 632. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-115. - 30.9i)T + (1.22e6 + 7.09e5i)T^{2} \) |
| 19 | \( 1 + (-303. + 525. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (428. + 1.59e3i)T + (-5.57e6 + 3.21e6i)T^{2} \) |
| 29 | \( 1 + 1.14e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (3.02e3 - 1.74e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (465. - 124. i)T + (6.00e7 - 3.46e7i)T^{2} \) |
| 41 | \( 1 - 1.54e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-7.87e3 + 7.87e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (6.68e3 + 2.49e4i)T + (-1.98e8 + 1.14e8i)T^{2} \) |
| 53 | \( 1 + (8.83e3 + 2.36e3i)T + (3.62e8 + 2.09e8i)T^{2} \) |
| 59 | \( 1 + (-3.47e3 - 6.02e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-7.78e3 - 4.49e3i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-5.17e3 + 1.93e4i)T + (-1.16e9 - 6.75e8i)T^{2} \) |
| 71 | \( 1 - 3.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (9.47e3 - 3.53e4i)T + (-1.79e9 - 1.03e9i)T^{2} \) |
| 79 | \( 1 + (8.10e3 + 4.68e3i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-7.47e3 - 7.47e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + (-1.66e4 + 2.88e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-6.11e4 + 6.11e4i)T - 8.58e9iT^{2} \) |
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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
−14.63407792034469993273994670799, −13.81083300895790490201853525260, −13.10486094260344888344166320900, −11.90534454129992021287998920451, −11.12208233839399680222666488711, −8.465668222454361765944060299855, −6.70955652826778936156528538052, −5.87250370080192960445878439770, −3.23892977602292910013042817934, −2.13767090134519880097578650988,
3.53247639834677998803572172182, 4.48230621564541221943704593409, 5.51727673007014803489307618709, 7.78920052569755459034098417996, 9.585668700698187158590529429556, 10.84435634069616109916411419982, 12.61003209769990981792595142770, 13.52407338748132979487622410075, 14.66242565291728104793689438948, 15.78068672348371786604377788730
