| L(s) = 1 | + (−1.14 + 1.64i)2-s + (−1.35 − 1.07i)3-s + (−1.38 − 3.75i)4-s + (−3.01 − 1.45i)5-s + (3.32 − 0.986i)6-s + (4.70 + 3.74i)7-s + (7.74 + 2.02i)8-s + (0.667 + 2.92i)9-s + (5.83 − 3.28i)10-s + (0.790 + 0.180i)11-s + (−2.17 + 6.57i)12-s + (−1.20 + 5.27i)13-s + (−11.5 + 3.42i)14-s + (2.51 + 5.22i)15-s + (−12.1 + 10.3i)16-s − 12.3·17-s + ⋯ |
| L(s) = 1 | + (−0.571 + 0.820i)2-s + (−0.451 − 0.359i)3-s + (−0.346 − 0.938i)4-s + (−0.603 − 0.290i)5-s + (0.553 − 0.164i)6-s + (0.671 + 0.535i)7-s + (0.967 + 0.252i)8-s + (0.0741 + 0.324i)9-s + (0.583 − 0.328i)10-s + (0.0718 + 0.0163i)11-s + (−0.181 + 0.548i)12-s + (−0.0926 + 0.405i)13-s + (−0.823 + 0.244i)14-s + (0.167 + 0.348i)15-s + (−0.760 + 0.649i)16-s − 0.727·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0998 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0998 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.293650 - 0.324603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293650 - 0.324603i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.14 - 1.64i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 29 | \( 1 + (27.4 + 9.24i)T \) |
| good | 5 | \( 1 + (3.01 + 1.45i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-4.70 - 3.74i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-0.790 - 0.180i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (1.20 - 5.27i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 + (-17.1 + 13.6i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (11.3 + 23.5i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-4.47 + 9.30i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (8.16 + 35.7i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 + 62.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-19.6 - 40.8i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (13.2 + 3.01i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-6.51 - 3.13i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 57.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (-63.2 + 79.2i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (42.6 - 9.72i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (73.2 + 16.7i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (15.6 - 7.51i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (50.3 - 11.5i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (14.4 - 11.5i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (24.2 + 11.6i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (59.5 + 74.6i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.20338993161006251101853958109, −9.953852397440575535015816125573, −8.899581707588716358351532703384, −8.164111367730510330103301898984, −7.28598934607685836807943570437, −6.32363671222407546296452821912, −5.23122095186921208585756102080, −4.35701128828176150134350743300, −1.96730569970835976704282767977, −0.27034863918150602103653972929,
1.47510040596308890190392441778, 3.32707057150292543216115034566, 4.19533690659595623212889973390, 5.40852390115222845405338001615, 7.14106900944769870431103261747, 7.80335317410082445337749066569, 8.861911070215987892991290079597, 9.974949562860362530270547153880, 10.61016498919092380099245343344, 11.60939347453429292232170968257
