L(s) = 1 | + (1.12 + 1.94i)2-s + (0.514 − 1.91i)3-s + (−1.53 + 2.65i)4-s + (−2.22 + 0.247i)5-s + (4.31 − 1.15i)6-s + (−1.10 − 0.638i)7-s − 2.39·8-s + (−0.820 − 0.473i)9-s + (−2.98 − 4.05i)10-s + (−5.27 − 1.41i)11-s + (4.30 + 4.30i)12-s + (3.50 + 0.840i)13-s − 2.87i·14-s + (−0.666 + 4.39i)15-s + (0.365 + 0.633i)16-s + (3.11 − 0.833i)17-s + ⋯ |
L(s) = 1 | + (0.795 + 1.37i)2-s + (0.296 − 1.10i)3-s + (−0.766 + 1.32i)4-s + (−0.993 + 0.110i)5-s + (1.76 − 0.472i)6-s + (−0.418 − 0.241i)7-s − 0.848·8-s + (−0.273 − 0.157i)9-s + (−0.943 − 1.28i)10-s + (−1.59 − 0.426i)11-s + (1.24 + 1.24i)12-s + (0.972 + 0.233i)13-s − 0.768i·14-s + (−0.172 + 1.13i)15-s + (0.0913 + 0.158i)16-s + (0.754 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05209 + 0.528890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05209 + 0.528890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.22 - 0.247i)T \) |
| 13 | \( 1 + (-3.50 - 0.840i)T \) |
good | 2 | \( 1 + (-1.12 - 1.94i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.514 + 1.91i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (1.10 + 0.638i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.27 + 1.41i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-3.11 + 0.833i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.315 - 1.17i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.160 - 0.0428i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (8.41 - 4.85i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.233 - 0.233i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.14 - 0.660i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.129 + 0.483i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.72 + 6.43i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 3.20iT - 47T^{2} \) |
| 53 | \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.00222 - 0.000595i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.695 - 1.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.03 + 5.26i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.7 + 3.14i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 - 11.1iT - 79T^{2} \) |
| 83 | \( 1 - 2.65iT - 83T^{2} \) |
| 89 | \( 1 + (-1.86 + 6.96i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (2.09 - 3.62i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.11609218214936770955610446714, −13.87218223261271993366214079877, −13.20594260526986949477242247964, −12.38131849067684577762253955255, −10.74987865572969365470378114257, −8.373273502107031791891166909769, −7.65246359536913134567674798152, −6.83213068706262333605999386208, −5.40770898675932553382495091950, −3.56328042452326942346974060767,
3.06403938626189378884197003228, 4.05877097203690452316556916419, 5.27622605467119284392717744743, 7.902858549223609046143390497729, 9.503762359803084872668407745640, 10.48844277377549798780080376334, 11.27364515380475546288617620923, 12.55975698639598009649931313858, 13.25858309978173176049927378845, 14.79842839305128995833780538080