| L(s) = 1 | + (0.561 − 0.827i)2-s + (1.73 − 0.0579i)3-s + (−0.370 − 0.928i)4-s + (0.235 − 0.509i)5-s + (0.923 − 1.46i)6-s + (−0.907 − 3.26i)7-s + (−0.976 − 0.214i)8-s + (2.99 − 0.200i)9-s + (−0.289 − 0.480i)10-s + (−3.58 − 3.39i)11-s + (−0.694 − 1.58i)12-s + (−0.641 + 5.89i)13-s + (−3.21 − 1.08i)14-s + (0.378 − 0.895i)15-s + (−0.725 + 0.687i)16-s + (4.82 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.396 − 0.585i)2-s + (0.999 − 0.0334i)3-s + (−0.185 − 0.464i)4-s + (0.105 − 0.227i)5-s + (0.377 − 0.598i)6-s + (−0.342 − 1.23i)7-s + (−0.345 − 0.0760i)8-s + (0.997 − 0.0668i)9-s + (−0.0914 − 0.152i)10-s + (−1.08 − 1.02i)11-s + (−0.200 − 0.458i)12-s + (−0.177 + 1.63i)13-s + (−0.858 − 0.289i)14-s + (0.0976 − 0.231i)15-s + (−0.181 + 0.171i)16-s + (1.17 + 0.325i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57141 - 1.39472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57141 - 1.39472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.561 + 0.827i)T \) |
| 3 | \( 1 + (-1.73 + 0.0579i)T \) |
| 59 | \( 1 + (-1.62 - 7.50i)T \) |
| good | 5 | \( 1 + (-0.235 + 0.509i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.907 + 3.26i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (3.58 + 3.39i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.641 - 5.89i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (-4.82 - 1.34i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (-2.18 - 1.66i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.378 + 0.713i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-2.22 + 1.50i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (1.61 + 2.12i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-2.49 - 11.3i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (10.3 + 5.49i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (-4.16 - 4.39i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (6.25 - 2.89i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-3.18 + 5.29i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (8.09 + 5.48i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 10.1i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (1.79 + 3.87i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (0.460 - 1.36i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (0.757 - 13.9i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (0.950 - 5.79i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (1.75 + 2.58i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (3.31 + 9.83i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−11.20839161977480908547131418229, −10.20039504174633638397261617906, −9.659353341654744808144592809133, −8.505536789358159713381656570089, −7.58769587402280476014820027444, −6.50777458897561382112667824446, −4.99487290471720927083820155265, −3.86098875439223449077976718291, −3.01926740003340855585489600431, −1.37977119500474326617026393153,
2.54281326354299733703572127236, 3.20788924893608243686357091784, 4.94742787749883846364933492810, 5.68061782084372214451703540236, 7.15819938921036787867813636898, 7.84540133478876098337813628399, 8.716583935271537453235668545500, 9.763789267586068646337204981090, 10.42592336569529651997434282603, 12.21413918389767419478550104007
