L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 1.41·6-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)12-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (0.707 − 1.22i)19-s + (1.22 + 0.707i)23-s + 0.999i·27-s + (0.707 + 1.22i)31-s + (1.22 − 0.707i)32-s + 0.999·36-s + (1.73 − 0.999i)38-s + ⋯ |
L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s − 1.41·6-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)12-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (0.707 − 1.22i)19-s + (1.22 + 0.707i)23-s + 0.999i·27-s + (0.707 + 1.22i)31-s + (1.22 − 0.707i)32-s + 0.999·36-s + (1.73 − 0.999i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.950339694\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.950339694\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.926726042189955947592919412175, −7.69901512224470038991668993902, −6.94492028934407086081256670257, −6.52620535441610807564477718768, −5.65442577967675461031713908475, −4.98360489738100685410801906590, −4.66326067555917149462612004000, −3.59818341518640792518916807816, −2.95358258536124268859835531527, −1.07686141981915975369219622955,
1.18090924044932987349559844839, 2.19911476048663858068875540649, 3.11115047923655715818064812853, 4.08941750954252452125694389242, 4.78610940781188336035139647781, 5.49646999661577311773092131907, 6.07939395807968519555104976083, 6.85140210238901812862380219067, 7.76411305667793230123634270999, 8.406843346066080927369138770288