| L(s) = 1 | + (−0.191 + 3.53i)2-s + (−0.887 + 5.11i)3-s + (−4.49 − 0.489i)4-s + (2.01 − 5.98i)5-s + (−17.9 − 4.11i)6-s + (16.3 + 24.1i)7-s + (−1.98 + 12.1i)8-s + (−25.4 − 9.08i)9-s + (20.7 + 8.27i)10-s + (−40.8 + 8.99i)11-s + (6.49 − 22.5i)12-s + (8.51 − 7.23i)13-s + (−88.2 + 53.1i)14-s + (28.8 + 15.6i)15-s + (−77.8 − 17.1i)16-s + (−23.1 − 15.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.0677 + 1.24i)2-s + (−0.170 + 0.985i)3-s + (−0.562 − 0.0611i)4-s + (0.180 − 0.535i)5-s + (−1.21 − 0.280i)6-s + (0.882 + 1.30i)7-s + (−0.0879 + 0.536i)8-s + (−0.941 − 0.336i)9-s + (0.656 + 0.261i)10-s + (−1.12 + 0.246i)11-s + (0.156 − 0.543i)12-s + (0.181 − 0.154i)13-s + (−1.68 + 1.01i)14-s + (0.496 + 0.269i)15-s + (−1.21 − 0.267i)16-s + (−0.330 − 0.223i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.505170 - 1.23688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.505170 - 1.23688i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.887 - 5.11i)T \) |
| 59 | \( 1 + (-433. + 133. i)T \) |
| good | 2 | \( 1 + (0.191 - 3.53i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (-2.01 + 5.98i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (-16.3 - 24.1i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (40.8 - 8.99i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (-8.51 + 7.23i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (23.1 + 15.6i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (26.4 - 49.8i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (-54.6 + 51.7i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (29.1 - 1.58i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (-152. + 80.5i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-139. + 22.9i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (214. - 226. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-6.71 + 30.5i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (242. - 81.6i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (344. - 137. i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (392. + 21.2i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (-308. - 50.6i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-55.9 - 165. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (-452. - 751. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (-720. - 333. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (55.4 + 199. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (-25.0 - 461. i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (732. - 1.21e3i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.93026191984864512156701467730, −11.72864762677163541983818425211, −10.87971908459693847964098111804, −9.517793421973041988483480868521, −8.542176681340713385340682584846, −8.020156355007903527209438620495, −6.28170359451940848476304845389, −5.27429366727052012601959140597, −4.82139768770868230564321624022, −2.52879896578694351778463274844,
0.61488685708920123180671883009, 1.91353181004576625347493850678, 3.13901365186272696422984448953, 4.77815581037435142441097823341, 6.53069613678404130074043310097, 7.40398823722418012370208126509, 8.504463459807977107983753295298, 10.15666814499641996484512681820, 10.93992237631678503082853812644, 11.29979801842578220177451683062
