| L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.146 − 0.828i)3-s + (0.766 − 0.642i)4-s + (−1.03 − 0.867i)5-s + (0.146 + 0.828i)6-s + (0.273 + 0.473i)7-s + (−0.500 + 0.866i)8-s + (2.15 + 0.784i)9-s + (1.26 + 0.461i)10-s + (0.5 − 0.866i)11-s + (−0.420 − 0.728i)12-s + (−0.306 − 1.73i)13-s + (−0.419 − 0.351i)14-s + (−0.869 + 0.729i)15-s + (0.173 − 0.984i)16-s + (2.21 − 0.805i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.0843 − 0.478i)3-s + (0.383 − 0.321i)4-s + (−0.462 − 0.387i)5-s + (0.0596 + 0.338i)6-s + (0.103 + 0.179i)7-s + (−0.176 + 0.306i)8-s + (0.718 + 0.261i)9-s + (0.400 + 0.145i)10-s + (0.150 − 0.261i)11-s + (−0.121 − 0.210i)12-s + (−0.0848 − 0.481i)13-s + (−0.112 − 0.0940i)14-s + (−0.224 + 0.188i)15-s + (0.0434 − 0.246i)16-s + (0.536 − 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.846364 - 0.504267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846364 - 0.504267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-1.75 + 3.98i)T \) |
| good | 3 | \( 1 + (-0.146 + 0.828i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.03 + 0.867i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.273 - 0.473i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (0.306 + 1.73i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 0.805i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.494 + 0.415i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.97 + 2.53i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.91 + 8.51i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + (1.97 - 11.1i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.16 - 6.01i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (3.71 + 1.35i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-7.92 + 6.65i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (6.42 - 2.33i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 1.12i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 3.79i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (9.21 + 7.72i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.58 - 8.96i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.187 - 1.06i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.64 - 4.58i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.77 - 10.0i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (11.7 - 4.25i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.14723013190574799634156157388, −9.912113276064458463004354719584, −9.254442681154404271168184552406, −7.972141357328789478433573378403, −7.70497446837181463018356587678, −6.54703190445587733880234870602, −5.42067863274900141243645534925, −4.15893754657957829194895129013, −2.46486634707466903171928058918, −0.865019881868517977515293037731,
1.58386678751046612768633342737, 3.37849107627237073087533712561, 4.15814047822714448914227664447, 5.65642712765670502424053505909, 7.16219989918194505053951912621, 7.49456197304200413587008809705, 8.882817810801636822374159516951, 9.558974421676732195324510217204, 10.45377889658670560078547286392, 11.08889975235058981136526134475
