L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s + 0.999i·8-s + (1.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s − i·29-s + (−1.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)40-s + (0.866 + 0.499i)44-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s + 0.999i·8-s + (1.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s − i·29-s + (−1.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)40-s + (0.866 + 0.499i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.561816920\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.561816920\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−8.804970505731845945680269475825, −8.043639191691537781513764962611, −7.15014460247115465133257065028, −6.22331809369899560425638436067, −5.73354281641219609440268816672, −5.03751937344618109071226814587, −4.31656233748741455226282786416, −3.52796638331368601987844280031, −2.27172396268634410937145333381, −1.30068320453542783647605866453,
1.65469086099711228253362739435, 2.29776339363999742878148785697, 3.28503536118662693449654919080, 3.84982242891029496866036603075, 4.98631094857944052906232788493, 5.75898325855555291322067703194, 6.51115755581602765917300600623, 6.89337996629614449355205227455, 7.67906435256990149148093352557, 9.152677016315313831018395252183