| L(s) = 1 | + (−0.848 − 1.32i)2-s + (0.567 + 1.63i)3-s + (−0.192 + 0.421i)4-s + (1.50 − 0.442i)5-s + (1.67 − 2.13i)6-s + (1.83 − 1.59i)7-s + (−2.38 + 0.343i)8-s + (−2.35 + 1.85i)9-s + (−1.86 − 1.61i)10-s + (−0.834 − 0.536i)11-s + (−0.798 − 0.0755i)12-s + (−1.51 + 1.75i)13-s + (−3.65 − 1.07i)14-s + (1.57 + 2.21i)15-s + (3.08 + 3.56i)16-s + (1.86 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.600 − 0.933i)2-s + (0.327 + 0.944i)3-s + (−0.0962 + 0.210i)4-s + (0.673 − 0.197i)5-s + (0.685 − 0.872i)6-s + (0.693 − 0.601i)7-s + (−0.844 + 0.121i)8-s + (−0.784 + 0.619i)9-s + (−0.588 − 0.510i)10-s + (−0.251 − 0.161i)11-s + (−0.230 − 0.0218i)12-s + (−0.420 + 0.485i)13-s + (−0.977 − 0.286i)14-s + (0.407 + 0.571i)15-s + (0.771 + 0.890i)16-s + (0.451 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.769853 - 0.281183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769853 - 0.281183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.567 - 1.63i)T \) |
| 23 | \( 1 + (4.71 - 0.883i)T \) |
| good | 2 | \( 1 + (0.848 + 1.32i)T + (-0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.50 + 0.442i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.83 + 1.59i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (0.834 + 0.536i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.51 - 1.75i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.86 - 4.07i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (5.75 + 2.62i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-7.93 + 3.62i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.409 + 2.84i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.564 - 1.92i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.89 + 9.84i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-10.8 - 1.56i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 0.223iT - 47T^{2} \) |
| 53 | \( 1 + (-7.41 - 8.55i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.44 + 1.25i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-4.00 + 0.576i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (0.684 + 1.06i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.546 + 0.850i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.52 + 7.71i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.51 + 4.77i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (0.661 + 0.194i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.253 - 1.76i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.95 - 10.0i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.59710367472905420873632864779, −13.66490583928152655165578734934, −12.08122063986677565699406414831, −10.81627695523622865139112893578, −10.27230726748804224154311856305, −9.237505370972691738375923773967, −8.168535384202320752643784396595, −5.87559028628377601706419522245, −4.21586186065228816895365197809, −2.21000934796227100838784976204,
2.50086612926162151074805697013, 5.60855933727515572180548746528, 6.69821587312343128463501347300, 7.912464673406863176242269818890, 8.638871270397441923782790256100, 10.00894916239136768864099909437, 11.84877500214426925701707544669, 12.66518152329530935585129444942, 14.15207498044545269168447520320, 14.76546629156176850549530349255
