L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s + i·17-s + 0.999i·18-s + 19-s + (0.866 − 0.499i)22-s + (0.5 − 0.866i)24-s − 0.999i·27-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s + i·17-s + 0.999i·18-s + 19-s + (0.866 − 0.499i)22-s + (0.5 − 0.866i)24-s − 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.499945022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499945022\) |
\(L(1)\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.537900266645637676292854239086, −8.447362048204654843682769982276, −7.77345728793864853762069993015, −7.00719014879069502491016080924, −6.12290699527751651355704351374, −5.79062229728706564662763946412, −4.79955764166736163601319584662, −3.92656340199539342136407000614, −2.86178845469027986955601962754, −1.45206446334580220141022558647,
1.13118534805815232843963117209, 2.54040636712250824696837986913, 3.71925903968202968328448534812, 4.41729268976137092322007361574, 5.25041359226669070510417715945, 5.81392140916424762054633402591, 6.93449441084823619800643345840, 7.27591336277501849364418465696, 8.954361058897877069858290353112, 9.728758128060513402488796509944