| L(s) = 1 | + (1.04 − 0.955i)2-s + (1.25 + 2.75i)3-s + (0.172 − 1.99i)4-s + (−0.794 − 0.916i)5-s + (3.94 + 1.66i)6-s + (2.19 + 1.40i)7-s + (−1.72 − 2.24i)8-s + (−4.03 + 4.65i)9-s + (−1.70 − 0.196i)10-s + (−1.15 + 0.165i)11-s + (5.70 − 2.02i)12-s + (−2.87 − 4.47i)13-s + (3.63 − 0.627i)14-s + (1.52 − 3.33i)15-s + (−3.94 − 0.688i)16-s + (1.35 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.737 − 0.675i)2-s + (0.725 + 1.58i)3-s + (0.0863 − 0.996i)4-s + (−0.355 − 0.409i)5-s + (1.60 + 0.680i)6-s + (0.828 + 0.532i)7-s + (−0.609 − 0.792i)8-s + (−1.34 + 1.55i)9-s + (−0.538 − 0.0620i)10-s + (−0.347 + 0.0499i)11-s + (1.64 − 0.585i)12-s + (−0.798 − 1.24i)13-s + (0.970 − 0.167i)14-s + (0.393 − 0.861i)15-s + (−0.985 − 0.172i)16-s + (0.327 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93714 + 0.0753432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93714 + 0.0753432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 + 0.955i)T \) |
| 23 | \( 1 + (-0.715 - 4.74i)T \) |
| good | 3 | \( 1 + (-1.25 - 2.75i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.794 + 0.916i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.19 - 1.40i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (1.15 - 0.165i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.87 + 4.47i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.35 - 4.60i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 5.58i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.739 + 2.51i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (6.33 + 2.89i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (5.01 - 5.79i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.14 - 1.32i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.698 + 0.319i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 1.52iT - 47T^{2} \) |
| 53 | \( 1 + (-7.13 - 4.58i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.213 + 0.137i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.27 - 7.18i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-4.29 - 0.618i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-12.6 - 1.81i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-7.02 - 2.06i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-8.57 + 5.50i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.11 - 2.70i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (9.40 - 4.29i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.16 + 7.07i)T + (13.8 - 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.60912699849680711668235822393, −11.50960754750126568737989311544, −10.64985409056501226702030676811, −9.852738011342848023392077429403, −8.880627666091497884774381321852, −7.87491148862102263699122404154, −5.44436287156834631913110760234, −4.90045989608138254147580906154, −3.75043465482734241639346053990, −2.56009296637436716561941791808,
2.11571873335099078443286795048, 3.56725710872584888768198910161, 5.19345828894195896305636671559, 6.79085830259890727180992124507, 7.33080599997170156851566765481, 7.976767458165593989110186975583, 9.105357720493066993132730862627, 11.12712790936779261615715270884, 12.07831198636695426124240512704, 12.66727395795408616426217426804
