L(s) = 1 | + (1.29 − 2.23i)3-s + (0.5 + 0.866i)5-s + (−1.83 − 3.18i)9-s + (−0.839 + 1.45i)11-s + 4.84·13-s + 2.58·15-s + (1 − 1.73i)17-s + (3.42 + 5.92i)19-s + (1.13 + 1.95i)23-s + (−0.499 + 0.866i)25-s − 1.75·27-s + 3.32·29-s + (4.58 − 7.94i)31-s + (2.16 + 3.75i)33-s + (1.42 + 2.46i)37-s + ⋯ |
L(s) = 1 | + (0.746 − 1.29i)3-s + (0.223 + 0.387i)5-s + (−0.613 − 1.06i)9-s + (−0.253 + 0.438i)11-s + 1.34·13-s + 0.667·15-s + (0.242 − 0.420i)17-s + (0.785 + 1.36i)19-s + (0.235 + 0.408i)23-s + (−0.0999 + 0.173i)25-s − 0.337·27-s + 0.616·29-s + (0.823 − 1.42i)31-s + (0.377 + 0.653i)33-s + (0.233 + 0.405i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.606920492\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606920492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.29 + 2.23i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.839 - 1.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.42 - 5.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.13 - 1.95i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + (-4.58 + 7.94i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.42 - 2.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.52T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + (6.10 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.74 - 6.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.24 + 5.62i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.87 + 4.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (5.84 - 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.84 + 4.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + (2.92 + 5.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.887212308872020648565565107102, −7.979787523652368687284543803565, −7.74723344792843669507407570720, −6.70296564184533675174050377282, −6.20126455560215396979961995596, −5.19934938951959715654996334085, −3.77954865524536929482978870565, −3.00127615710376660319121463950, −1.98450117000370584814677719430, −1.11354062419545433122464551097,
1.16963977662724370370158520726, 2.81406393441930216580889252527, 3.38135196367037019280944439475, 4.40359747244801973447943750512, 5.02321695872046806588165510405, 5.96904161205220414073053815316, 6.91737512514939515784365315121, 8.208418567259846324735784569332, 8.608148263433659798544342935840, 9.215706138470232805701654789428