L(s) = 1 | + (1.23 + 3.80i)2-s + (8.60 − 6.25i)3-s + (−12.9 + 9.40i)4-s + (−38.0 + 40.9i)5-s + (34.4 + 25.0i)6-s − 106.·7-s + (−51.7 − 37.6i)8-s + (−40.1 + 123. i)9-s + (−202. − 93.9i)10-s + (155. + 479. i)11-s + (−52.5 + 161. i)12-s + (−61.5 + 189. i)13-s + (−131. − 405. i)14-s + (−70.9 + 590. i)15-s + (79.1 − 243. i)16-s + (−415. − 302. i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.552 − 0.401i)3-s + (−0.404 + 0.293i)4-s + (−0.680 + 0.733i)5-s + (0.390 + 0.283i)6-s − 0.822·7-s + (−0.286 − 0.207i)8-s + (−0.165 + 0.508i)9-s + (−0.641 − 0.297i)10-s + (0.388 + 1.19i)11-s + (−0.105 + 0.324i)12-s + (−0.101 + 0.311i)13-s + (−0.179 − 0.553i)14-s + (−0.0814 + 0.677i)15-s + (0.0772 − 0.237i)16-s + (−0.348 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.294074 + 1.17484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.294074 + 1.17484i\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + (38.0 - 40.9i)T \) |
good | 3 | \( 1 + (-8.60 + 6.25i)T + (75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-155. - 479. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (61.5 - 189. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (415. + 302. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.27e3 + 923. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-418. - 1.28e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.91e3 + 2.84e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.82e3 - 4.23e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-43.3 + 133. i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.46e3 - 1.06e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 6.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.00e4 + 7.26e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.24e4 + 1.62e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.33e4 - 4.09e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.18e3 - 1.28e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (4.95e4 + 3.59e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.16e4 - 8.47e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.49e4 - 7.69e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (6.75e4 - 4.90e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.80e4 + 2.03e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.70e4 + 5.26e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.47e4 + 2.52e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.04087845810234775725871702264, −14.01918859855524191035282423573, −12.99727656608798418777730628669, −11.77901051056052042177473186849, −10.14699847196536521148928757202, −8.684116618596388050511968825861, −7.36707280390930826700538220109, −6.59925365513201426212438563172, −4.42590193827876770518077243431, −2.71347297683121664282172392261,
0.53775318021160487944068892785, 3.13784345041066502496752835595, 4.21208733255306631652829406207, 6.14968795555482694376857894881, 8.391478023396951916234530692234, 9.147050102221988282630277211625, 10.50581498773276479619660755042, 11.88871244804013143790272176930, 12.77960479225636864749235932219, 13.95752725075076393714391967149