| L(s) = 1 | + (−6.81 − 2.10i)2-s + (3.52 − 8.98i)3-s + (15.5 + 10.6i)4-s + (80.2 − 12.0i)5-s + (−42.9 + 53.8i)6-s + (−85.3 + 97.5i)7-s + (58.3 + 73.2i)8-s + (109. + 101. i)9-s + (−572. − 86.2i)10-s + (415. − 385. i)11-s + (150. − 102. i)12-s + (20.6 + 90.5i)13-s + (786. − 485. i)14-s + (174. − 763. i)15-s + (−464. − 1.18e3i)16-s + (−5.71 − 76.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.20 − 0.371i)2-s + (0.226 − 0.576i)3-s + (0.487 + 0.332i)4-s + (1.43 − 0.216i)5-s + (−0.486 + 0.610i)6-s + (−0.658 + 0.752i)7-s + (0.322 + 0.404i)8-s + (0.452 + 0.419i)9-s + (−1.81 − 0.272i)10-s + (1.03 − 0.961i)11-s + (0.301 − 0.205i)12-s + (0.0339 + 0.148i)13-s + (1.07 − 0.662i)14-s + (0.199 − 0.876i)15-s + (−0.453 − 1.15i)16-s + (−0.00480 − 0.0640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.03175 - 0.629006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03175 - 0.629006i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (85.3 - 97.5i)T \) |
| good | 2 | \( 1 + (6.81 + 2.10i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (-3.52 + 8.98i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (-80.2 + 12.0i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (-415. + 385. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-20.6 - 90.5i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (5.71 + 76.3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-1.25e3 + 2.17e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (51.6 - 688. i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (5.08e3 + 2.44e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-3.35e3 - 5.81e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.42e3 - 3.70e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (6.18e3 + 7.75e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-1.26e4 + 1.59e4i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-2.64e4 - 8.14e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (-9.74e3 - 6.64e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-772. - 116. i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (8.39e3 - 5.72e3i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-1.77e4 - 3.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (7.59e4 - 3.65e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-2.96e3 + 915. i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-3.55e4 + 6.15e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (6.67e3 - 2.92e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-2.50e4 - 2.32e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + 1.49e4T + 8.58e9T^{2} \) |
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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
−13.92274202984052550002412191414, −13.45251216233722796060576400365, −11.92585590564180699684439246896, −10.47730645994847030592896806255, −9.299918447144181434523750764214, −8.827589950428086613395987593493, −7.00531087601936105775652735769, −5.55118257458697341902742877620, −2.42254152244723579550291360809, −1.13827723774210951766815213662,
1.39195651121882125395974784293, 3.95390124495980555456368792798, 6.26523764149330729021163442185, 7.35219772705841496839439015255, 9.259523545015295441130437148406, 9.742328560162704864703724388427, 10.40000102931515129069053644997, 12.61928210611627691211107483991, 13.82513462335979505827002686920, 14.95012287563337976863800466727
