| L(s) = 1 | + (0.104 + 0.994i)2-s + (1.67 − 0.435i)3-s + (−0.978 + 0.207i)4-s + (0.789 + 1.77i)5-s + (0.608 + 1.62i)6-s + (−2.28 − 1.32i)7-s + (−0.309 − 0.951i)8-s + (2.62 − 1.45i)9-s + (−1.68 + 0.970i)10-s + (3.21 − 0.795i)11-s + (−1.54 + 0.774i)12-s + (3.98 + 5.48i)13-s + (1.07 − 2.41i)14-s + (2.09 + 2.62i)15-s + (0.913 − 0.406i)16-s + (−0.459 + 4.36i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (0.967 − 0.251i)3-s + (−0.489 + 0.103i)4-s + (0.353 + 0.792i)5-s + (0.248 + 0.662i)6-s + (−0.865 − 0.501i)7-s + (−0.109 − 0.336i)8-s + (0.873 − 0.486i)9-s + (−0.531 + 0.306i)10-s + (0.970 − 0.239i)11-s + (−0.447 + 0.223i)12-s + (1.10 + 1.52i)13-s + (0.288 − 0.645i)14-s + (0.541 + 0.678i)15-s + (0.228 − 0.101i)16-s + (−0.111 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66616 + 1.08149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66616 + 1.08149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (-1.67 + 0.435i)T \) |
| 7 | \( 1 + (2.28 + 1.32i)T \) |
| 11 | \( 1 + (-3.21 + 0.795i)T \) |
| good | 5 | \( 1 + (-0.789 - 1.77i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 5.48i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.459 - 4.36i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.0977 - 0.459i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.703 + 0.406i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.927 - 2.85i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (7.66 + 3.41i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (6.60 + 7.33i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.04 - 3.22i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.53iT - 43T^{2} \) |
| 47 | \( 1 + (-1.94 + 9.13i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 3.82i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (0.627 + 2.95i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (2.08 + 4.67i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (1.96 + 3.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.37 - 11.5i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.63 + 7.69i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (7.45 - 0.784i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.29 - 3.12i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 6.90i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.56 + 1.13i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.04696269849239857613648689508, −10.10419379771533747747530453037, −9.111671490400930466413501022217, −8.679937505917839417958656171753, −7.27784154414034718225900995387, −6.67528363786752227954802844409, −6.08281219947666874112291347639, −3.97479541498090374650777078440, −3.59303871104014439780108082184, −1.81949115945014230997392687185,
1.33844760791606464659359117088, 2.86672678337825391021990820294, 3.67309793590331895483088837336, 4.93773776794980816816881608209, 5.99750800763466530978526551407, 7.41552314521755367643416530546, 8.736391641413080993035700633475, 9.031923612878147152516300403873, 9.845402850890188377859223540954, 10.69933469195999674536209221915
