| 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
−14.47260102792744210529182701936, −13.91758909538892792823111160209, −12.07195934908415020691575983714, −11.16391857146136981797164762114, −9.008128118922282388056072537973, −8.717819545641382587905027358250, −7.20497069746294625405813607004, −5.36163169482100976711081617992, −4.06705419945723680538966939003, −1.06018097477037664498428525595,
1.66313416697470294501398931423, 3.34098289696810906773342860807, 5.21339138599580846137805838851, 7.66455853435144374142794352576, 8.113338423859876917939450926028, 10.09864091139713391416331140404, 11.06263320210793463563852882010, 11.94368558012092245762694170159, 13.45846600535916290891947975901, 14.49252401510012584655553894311
