| L(s) = 1 | + (−3.39 + 1.63i)2-s + (−13.4 − 16.8i)3-s + (−11.0 + 13.9i)4-s + (21.4 − 10.3i)5-s + (73.1 + 35.2i)6-s + (143. + 179. i)7-s + (41.7 − 182. i)8-s + (−49.1 + 215. i)9-s + (−56.0 + 70.3i)10-s + (58.8 + 257. i)11-s + 383.·12-s + (120. + 526. i)13-s + (−781. − 376. i)14-s + (−462. − 222. i)15-s + (30.7 + 134. i)16-s + 517.·17-s + ⋯ |
| L(s) = 1 | + (−0.600 + 0.289i)2-s + (−0.861 − 1.08i)3-s + (−0.346 + 0.434i)4-s + (0.384 − 0.185i)5-s + (0.829 + 0.399i)6-s + (1.10 + 1.38i)7-s + (0.230 − 1.01i)8-s + (−0.202 + 0.886i)9-s + (−0.177 + 0.222i)10-s + (0.146 + 0.642i)11-s + 0.768·12-s + (0.197 + 0.864i)13-s + (−1.06 − 0.513i)14-s + (−0.531 − 0.255i)15-s + (0.0300 + 0.131i)16-s + 0.434·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.736024 + 0.362902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736024 + 0.362902i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (498. - 4.50e3i)T \) |
| good | 2 | \( 1 + (3.39 - 1.63i)T + (19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (13.4 + 16.8i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-21.4 + 10.3i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (-143. - 179. i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-58.8 - 257. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-120. - 526. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 - 517.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-248. + 311. i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-1.32e3 - 640. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (-6.37e3 + 3.06e3i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (3.05e3 - 1.33e4i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.53e4 + 7.38e3i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (4.81e3 + 2.10e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-510. + 245. i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 - 2.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (1.04e4 + 1.31e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.08e4 + 4.73e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (4.97e3 + 2.17e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-2.65e4 - 1.27e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (3.90e3 - 1.71e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (2.83e4 - 3.55e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.74e4 + 3.24e4i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (1.07e5 - 1.34e5i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.83322910192163573428210918612, −15.19603560768760920726979223636, −13.52503719625653757303924161187, −12.29323835760044219120844199003, −11.59645374411159335256679669702, −9.385226452822508593149716092615, −8.161238012590413670388373238371, −6.77592480914456736442218725363, −5.16948919740266971611247733827, −1.57049333507687676796510743638,
0.815592766996718714344267633070, 4.39112272667043349079789228102, 5.67327991243849524587857754004, 8.089668544145771647931412118033, 9.882934288265971263650703597731, 10.61211150250110531306029011460, 11.32171047328693763499567821024, 13.71433870616176518701364513407, 14.65042354639954418102839075814, 16.22884872258990744994162110295
