L(s) = 1 | + (0.258 + 0.965i)3-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)11-s + (−0.707 + 0.707i)13-s − 15-s + (0.965 − 0.258i)17-s + (−1.22 + 0.707i)19-s + (−1.36 − 0.366i)23-s + (−0.866 − 0.499i)25-s + (0.707 + 0.707i)27-s − i·29-s + (−0.965 − 0.258i)33-s + (−0.366 + 1.36i)37-s + (−0.866 − 0.500i)39-s + (1 − i)43-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)3-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)11-s + (−0.707 + 0.707i)13-s − 15-s + (0.965 − 0.258i)17-s + (−1.22 + 0.707i)19-s + (−1.36 − 0.366i)23-s + (−0.866 − 0.499i)25-s + (0.707 + 0.707i)27-s − i·29-s + (−0.965 − 0.258i)33-s + (−0.366 + 1.36i)37-s + (−0.866 − 0.500i)39-s + (1 − i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9675722397\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9675722397\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
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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
−9.234005029997574889268616351610, −8.090864925854056322361018907254, −7.67376314684237361931678988074, −6.77069364529835214918688827313, −6.14958658945292607292836953427, −5.05879856006719447139549415845, −4.24306927178663531178180237693, −3.81972253575602270148493977590, −2.73868125658618933560953525388, −1.97681050069439424252177274782,
0.50100356431413836035387006312, 1.65664357809761840479719410472, 2.55707034508555427875102035174, 3.61551579642269161278477767307, 4.54865656825409873593977627410, 5.42977058959913834622141144113, 5.98921469155877033824623793685, 7.04032485346982996182117056955, 7.72337979136163799881028153710, 8.182286988555549980781113232985