| L(s) = 1 | + (−1.23 − 3.80i)2-s + (4.87 − 3.53i)3-s + (−12.9 + 9.40i)4-s + (−25.7 + 49.6i)5-s + (−19.4 − 14.1i)6-s + 192.·7-s + (51.7 + 37.6i)8-s + (−63.8 + 196. i)9-s + (220. + 36.6i)10-s + (102. + 316. i)11-s + (−29.7 + 91.6i)12-s + (307. − 947. i)13-s + (−237. − 732. i)14-s + (50.0 + 332. i)15-s + (79.1 − 243. i)16-s + (497. + 361. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.312 − 0.227i)3-s + (−0.404 + 0.293i)4-s + (−0.460 + 0.887i)5-s + (−0.220 − 0.160i)6-s + 1.48·7-s + (0.286 + 0.207i)8-s + (−0.262 + 0.809i)9-s + (0.697 + 0.115i)10-s + (0.255 + 0.787i)11-s + (−0.0596 + 0.183i)12-s + (0.505 − 1.55i)13-s + (−0.324 − 0.998i)14-s + (0.0574 + 0.382i)15-s + (0.0772 − 0.237i)16-s + (0.417 + 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0474i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.66453 + 0.0394791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66453 + 0.0394791i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 5 | \( 1 + (25.7 - 49.6i)T \) |
| good | 3 | \( 1 + (-4.87 + 3.53i)T + (75.0 - 231. i)T^{2} \) |
| 7 | \( 1 - 192.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-102. - 316. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-307. + 947. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-497. - 361. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.84e3 - 1.34e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-754. - 2.32e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.47e3 + 1.06e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (6.03e3 + 4.38e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.42e3 - 1.36e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.40e3 - 1.04e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 9.89e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.82e4 + 1.32e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-122. + 88.7i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-8.60e3 + 2.64e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.31e4 + 4.04e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-8.62e3 - 6.26e3i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.75e4 + 1.99e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.78e4 + 5.49e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.35e4 + 9.83e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.85e4 + 4.25e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.28e4 - 7.04e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.20e5 + 8.77e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.49252401510012584655553894311, −13.45846600535916290891947975901, −11.94368558012092245762694170159, −11.06263320210793463563852882010, −10.09864091139713391416331140404, −8.113338423859876917939450926028, −7.66455853435144374142794352576, −5.21339138599580846137805838851, −3.34098289696810906773342860807, −1.66313416697470294501398931423,
1.06018097477037664498428525595, 4.06705419945723680538966939003, 5.36163169482100976711081617992, 7.20497069746294625405813607004, 8.717819545641382587905027358250, 9.008128118922282388056072537973, 11.16391857146136981797164762114, 12.07195934908415020691575983714, 13.91758909538892792823111160209, 14.47260102792744210529182701936
