| L(s) = 1 | + (0.888 + 0.458i)2-s + (−2.06 − 1.62i)3-s + (0.580 + 0.814i)4-s + (2.01 + 1.92i)5-s + (−1.09 − 2.39i)6-s + (−1.50 + 2.17i)7-s + (0.142 + 0.989i)8-s + (0.922 + 3.80i)9-s + (0.911 + 2.63i)10-s + (5.31 − 2.74i)11-s + (0.125 − 2.62i)12-s + (−4.01 + 4.63i)13-s + (−2.33 + 1.24i)14-s + (−1.04 − 7.25i)15-s + (−0.327 + 0.945i)16-s + (5.04 + 0.481i)17-s + ⋯ |
| L(s) = 1 | + (0.628 + 0.324i)2-s + (−1.19 − 0.938i)3-s + (0.290 + 0.407i)4-s + (0.902 + 0.860i)5-s + (−0.445 − 0.976i)6-s + (−0.567 + 0.823i)7-s + (0.0503 + 0.349i)8-s + (0.307 + 1.26i)9-s + (0.288 + 0.832i)10-s + (1.60 − 0.826i)11-s + (0.0361 − 0.758i)12-s + (−1.11 + 1.28i)13-s + (−0.623 + 0.333i)14-s + (−0.269 − 1.87i)15-s + (−0.0817 + 0.236i)16-s + (1.22 + 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34207 + 0.549005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34207 + 0.549005i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 7 | \( 1 + (1.50 - 2.17i)T \) |
| 23 | \( 1 + (-4.54 - 1.53i)T \) |
| good | 3 | \( 1 + (2.06 + 1.62i)T + (0.707 + 2.91i)T^{2} \) |
| 5 | \( 1 + (-2.01 - 1.92i)T + (0.237 + 4.99i)T^{2} \) |
| 11 | \( 1 + (-5.31 + 2.74i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (4.01 - 4.63i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.04 - 0.481i)T + (16.6 + 3.21i)T^{2} \) |
| 19 | \( 1 + (-2.11 + 0.201i)T + (18.6 - 3.59i)T^{2} \) |
| 29 | \( 1 + (-0.0745 - 0.163i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (6.19 + 2.48i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.560 - 2.31i)T + (-32.8 + 16.9i)T^{2} \) |
| 41 | \( 1 + (3.49 - 1.02i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.637 + 4.43i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (5.59 + 9.68i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.78 - 1.50i)T + (49.2 - 19.6i)T^{2} \) |
| 59 | \( 1 + (-1.42 - 4.13i)T + (-46.3 + 36.4i)T^{2} \) |
| 61 | \( 1 + (-4.45 + 3.50i)T + (14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (0.0806 + 1.69i)T + (-66.6 + 6.36i)T^{2} \) |
| 71 | \( 1 + (-0.346 + 0.222i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.807 - 1.13i)T + (-23.8 + 68.9i)T^{2} \) |
| 79 | \( 1 + (4.32 + 0.833i)T + (73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (9.32 + 2.73i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-9.75 + 3.90i)T + (64.4 - 61.4i)T^{2} \) |
| 97 | \( 1 + (-15.5 + 4.56i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.76407410775352702004174832618, −11.37830282149302442218463701096, −9.923628045793820349926821519411, −9.028300060570037825754195947750, −7.23286153949167454521261396289, −6.64022840919730543393694563764, −6.02459170696522278057990408117, −5.23887745339199741371099181275, −3.33941566090845094808050245005, −1.81773578059284313570375124226,
1.11957773817304371231689396079, 3.44672688251766285982320141816, 4.67788867964144562668558589316, 5.26726736781237140090244807463, 6.20820893626442472774115394446, 7.33015498563169681343113596816, 9.448215963341456206710233117536, 9.748472039923629585516636687060, 10.51055749171491875224283941940, 11.56394584289736947388112998071
