| L(s) = 1 | + (−0.768 − 0.237i)2-s + (−8.72 + 22.2i)3-s + (−25.9 − 17.6i)4-s + (28.5 − 4.30i)5-s + (11.9 − 15.0i)6-s + (−107. + 71.7i)7-s + (31.7 + 39.8i)8-s + (−240. − 222. i)9-s + (−22.9 − 3.46i)10-s + (432. − 400. i)11-s + (618. − 421. i)12-s + (−206. − 904. i)13-s + (100. − 29.6i)14-s + (−153. + 672. i)15-s + (351. + 895. i)16-s + (−156. − 2.09e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 − 0.0419i)2-s + (−0.559 + 1.42i)3-s + (−0.809 − 0.551i)4-s + (0.510 − 0.0770i)5-s + (0.135 − 0.170i)6-s + (−0.832 + 0.553i)7-s + (0.175 + 0.220i)8-s + (−0.988 − 0.917i)9-s + (−0.0726 − 0.0109i)10-s + (1.07 − 0.998i)11-s + (1.24 − 0.845i)12-s + (−0.338 − 1.48i)13-s + (0.136 − 0.0403i)14-s + (−0.176 + 0.771i)15-s + (0.343 + 0.874i)16-s + (−0.131 − 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0549 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0549 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.282010 - 0.297970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282010 - 0.297970i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (107. - 71.7i)T \) |
| good | 2 | \( 1 + (0.768 + 0.237i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (8.72 - 22.2i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (-28.5 + 4.30i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (-432. + 400. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (206. + 904. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (156. + 2.09e3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (953. - 1.65e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (24.4 - 326. i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (-3.83e3 - 1.84e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (3.56e3 + 6.16e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.54e3 - 3.77e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (8.49e3 + 1.06e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (2.37e3 - 2.97e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.46e4 + 4.53e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (-3.38e3 - 2.30e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-3.52e4 - 5.31e3i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-1.24e4 + 8.46e3i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (2.45e4 + 4.24e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.52e4 - 1.21e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (5.66e4 - 1.74e4i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-779. + 1.34e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-824. + 3.61e3i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (1.80e4 + 1.67e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 - 3.71e4T + 8.58e9T^{2} \) |
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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
−14.47894674658156595875197290423, −13.29903890061593248906822402913, −11.71969051609266817913866801866, −10.31047371440068457604808899257, −9.686546443807507549240244902776, −8.777116352144559017284581113567, −5.97705877213865634283587550626, −5.20361301737259664672035939224, −3.55100159833810699119503314356, −0.24390201042765851654518118221,
1.67808477191005070143749659636, 4.23080505994607459204424958433, 6.46580606349733893252682319291, 7.05971101430290719808267328526, 8.744228891489194903309718356177, 9.986482948833307444146043740477, 11.83998589501374982734677370864, 12.71568750782748401720710174246, 13.42592190371598127809047883083, 14.45829454034823536846107797122
