| L(s) = 1 | + (0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.0415 + 0.395i)5-s + (0.809 + 0.587i)6-s + (−0.168 + 2.64i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.198 + 0.344i)10-s + (−2.37 + 2.31i)11-s + (0.499 + 0.866i)12-s + (−2.60 + 1.89i)13-s + (−1.22 + 2.34i)14-s + (−0.122 + 0.378i)15-s + (−0.104 + 0.994i)16-s + (5.72 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (0.564 + 0.120i)3-s + (0.334 + 0.371i)4-s + (−0.0185 + 0.176i)5-s + (0.330 + 0.239i)6-s + (−0.0636 + 0.997i)7-s + (0.109 + 0.336i)8-s + (0.304 + 0.135i)9-s + (−0.0628 + 0.108i)10-s + (−0.715 + 0.698i)11-s + (0.144 + 0.249i)12-s + (−0.722 + 0.525i)13-s + (−0.328 + 0.626i)14-s + (−0.0317 + 0.0976i)15-s + (−0.0261 + 0.248i)16-s + (1.38 − 0.617i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88599 + 1.36219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88599 + 1.36219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (0.168 - 2.64i)T \) |
| 11 | \( 1 + (2.37 - 2.31i)T \) |
| good | 5 | \( 1 + (0.0415 - 0.395i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (2.60 - 1.89i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.72 + 2.54i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.36 + 3.73i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (3.00 + 5.21i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.61 + 4.95i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.0330 - 0.314i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (3.19 - 0.679i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.86 - 8.80i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 + (-5.26 + 5.84i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (1.22 + 11.6i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (5.29 + 5.88i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.383 + 3.65i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (7.15 - 12.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.67 + 3.39i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.49 - 6.09i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (0.608 + 0.270i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-7.09 - 5.15i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.97 - 5.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.7 + 9.23i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.52281773600950090478402502769, −10.14661185657528103645403537888, −9.491913028164566439945826197809, −8.379691300344602520385377420804, −7.52153146261252138543087996026, −6.63538366690127775552401134237, −5.32637118042242043507400492197, −4.65973519212064578312431756277, −3.09804026268304598677271028362, −2.33734591703364908658692697943,
1.26054300150286481420203214997, 3.01854197065456701030933997999, 3.73760688043738292130233584335, 5.07771858815065567402275782500, 5.97041167648018599572242118399, 7.49244368039490437262514737453, 7.79224567140259269267897884248, 9.179739227529804979891906044865, 10.36698514859589190083720878388, 10.55500840125470953153519488049
