L(s) = 1 | + (0.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + 2·13-s − 1.99·15-s + (1 − 1.73i)17-s + (2 + 3.46i)19-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s + 6·29-s + (−4 + 6.92i)31-s + (−1.99 − 3.46i)33-s + (−3 − 5.19i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + 0.554·13-s − 0.516·15-s + (0.242 − 0.420i)17-s + (0.458 + 0.794i)19-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s + 1.11·29-s + (−0.718 + 1.24i)31-s + (−0.348 − 0.603i)33-s + (−0.493 − 0.854i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.630172600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630172600\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 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
−8.550507152483425934754051193297, −8.188078032661879302180944799750, −7.21648084118425622928209066022, −6.34364180046167945218035585768, −5.68151763230287155213017980195, −4.60967232176112934041742650917, −3.76257268075429955880624032348, −2.90921674254738151029893545694, −1.52133213432043857164100598109, −0.56446642265163695687474452757,
1.52673126472786782996170552848, 2.74419785092634319081542966376, 3.64207021248829800110982891933, 4.24968154057555195751315620520, 5.27315478908970611821742714149, 6.23470169547653679603146887279, 7.08625306741529305752404752371, 7.66403392034007199236951213633, 8.504551369367385211718407990303, 9.450089286200852409431001625586