| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−7.20 − 5.23i)3-s + (−12.9 − 9.40i)4-s + (−5.94 + 55.5i)5-s + (28.8 − 20.9i)6-s − 16.8·7-s + (51.7 − 37.6i)8-s + (−50.6 − 155. i)9-s + (−204. − 91.3i)10-s + (158. − 488. i)11-s + (44.0 + 135. i)12-s + (−161. − 496. i)13-s + (20.8 − 64.2i)14-s + (333. − 369. i)15-s + (79.1 + 243. i)16-s + (479. − 348. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.461 − 0.335i)3-s + (−0.404 − 0.293i)4-s + (−0.106 + 0.994i)5-s + (0.326 − 0.237i)6-s − 0.130·7-s + (0.286 − 0.207i)8-s + (−0.208 − 0.640i)9-s + (−0.645 − 0.288i)10-s + (0.395 − 1.21i)11-s + (0.0882 + 0.271i)12-s + (−0.264 − 0.814i)13-s + (0.0284 − 0.0876i)14-s + (0.382 − 0.423i)15-s + (0.0772 + 0.237i)16-s + (0.402 − 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.594998 - 0.415075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.594998 - 0.415075i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (5.94 - 55.5i)T \) |
| good | 3 | \( 1 + (7.20 + 5.23i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 16.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-158. + 488. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (161. + 496. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-479. + 348. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-498. + 362. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-442. + 1.36e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (6.39e3 + 4.64e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (460. - 334. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.90e3 - 8.95e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.94e3 + 9.07e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.20e3 - 5.23e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-8.25e3 - 6.00e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.40e3 + 7.39e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.69e4 + 5.22e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.69e4 - 2.68e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (4.65e4 + 3.37e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.72e4 + 5.29e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.94e4 - 5.04e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-3.80e4 + 2.76e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (9.80e3 - 3.01e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.73e4 - 1.26e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.52381283715534317678009949047, −13.39420765733986722552930124868, −11.87526074298131885625681421135, −10.82263405832107597596113789763, −9.427854445354395787652434552107, −7.892088447707191192249791710660, −6.63347912413147727335830739590, −5.70807345880502897859666407709, −3.33945343041481091513236465227, −0.43086841736861202433256998594,
1.71404964720070675862979370894, 4.16422879334655759926331690015, 5.33430955491523203056390201968, 7.52120254115942686617885013131, 9.054428271474494023507753239760, 9.978057503708619403687841889974, 11.37363628075747008919736432793, 12.27165133389479066302865957733, 13.29325887731256272557229134671, 14.72985103587053387862129610574
