| L(s) = 1 | + (−0.214 + 3.95i)2-s + (4.30 + 2.91i)3-s + (−7.64 − 0.831i)4-s + (−4.87 + 14.4i)5-s + (−12.4 + 16.3i)6-s + (8.97 + 13.2i)7-s + (−0.198 + 1.21i)8-s + (9.99 + 25.0i)9-s + (−56.2 − 22.3i)10-s + (49.8 − 10.9i)11-s + (−30.4 − 25.8i)12-s + (52.0 − 44.1i)13-s + (−54.2 + 32.6i)14-s + (−63.1 + 48.0i)15-s + (−64.8 − 14.2i)16-s + (−66.4 − 45.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.0758 + 1.39i)2-s + (0.827 + 0.561i)3-s + (−0.955 − 0.103i)4-s + (−0.436 + 1.29i)5-s + (−0.847 + 1.11i)6-s + (0.484 + 0.714i)7-s + (−0.00878 + 0.0536i)8-s + (0.370 + 0.928i)9-s + (−1.77 − 0.708i)10-s + (1.36 − 0.300i)11-s + (−0.732 − 0.622i)12-s + (1.10 − 0.942i)13-s + (−1.03 + 0.623i)14-s + (−1.08 + 0.826i)15-s + (−1.01 − 0.222i)16-s + (−0.948 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.317580 - 2.21520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317580 - 2.21520i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.30 - 2.91i)T \) |
| 59 | \( 1 + (-104. - 440. i)T \) |
| good | 2 | \( 1 + (0.214 - 3.95i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (4.87 - 14.4i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (-8.97 - 13.2i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (-49.8 + 10.9i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (-52.0 + 44.1i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (66.4 + 45.0i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (-22.2 + 41.9i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (-80.4 + 76.2i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (265. - 14.3i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (43.3 - 23.0i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-270. + 44.3i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (-77.4 + 81.8i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-20.5 + 93.4i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (-246. + 83.0i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (42.6 - 17.0i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (488. + 26.4i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (-741. - 121. i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-270. - 802. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (166. + 276. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (358. + 165. i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (-190. - 684. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (82.1 + 1.51e3i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (469. - 780. i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−13.31041308869890045991810083296, −11.37574013174838957864193826937, −10.92910956021460059862307466350, −9.191305640384447338492769858621, −8.642119931594549045661626930299, −7.54217799677613178834401604536, −6.70458081202577958783141824411, −5.52166710158806286546725287375, −3.99073656741209708366018957359, −2.65554284385426310567638473535,
1.11507633075267798770360391967, 1.69202721161374421812220659265, 3.83845729515153941447160436277, 4.20737378558021793699529165115, 6.56014304780814795566023940621, 7.87943209357720950226197036620, 9.133264942631750100214872746141, 9.284128707943542495975842643450, 11.08025948682590786341534182314, 11.66615327711456381793096496295
