L(s) = 1 | + (1 + 1.73i)2-s + (−0.999 + 1.73i)4-s + (−1 − 1.73i)5-s + (−2.5 − 0.866i)7-s + (1.99 − 3.46i)10-s + (−1 + 1.73i)11-s + 13-s + (−1.00 − 5.19i)14-s + (1.99 + 3.46i)16-s + (−0.5 − 0.866i)19-s + 3.99·20-s − 3.99·22-s + (0.500 − 0.866i)25-s + (1 + 1.73i)26-s + (3.99 − 3.46i)28-s − 4·29-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + (0.632 − 1.09i)10-s + (−0.301 + 0.522i)11-s + 0.277·13-s + (−0.267 − 1.38i)14-s + (0.499 + 0.866i)16-s + (−0.114 − 0.198i)19-s + 0.894·20-s − 0.852·22-s + (0.100 − 0.173i)25-s + (0.196 + 0.339i)26-s + (0.755 − 0.654i)28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.922996 + 0.613960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.922996 + 0.613960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−15.36560536854009396311158063456, −14.20505102814409591011598352646, −13.08004825465329500396874042276, −12.45412352413842447634937223453, −10.62667050589887060310804297029, −9.081367891683529429878694495088, −7.72619710623844035149662313582, −6.65467300645858647330142044042, −5.27089321981726813638725122212, −3.96149457595243024167394929240,
2.77247219492568790052505064510, 3.88826097130657345122151487621, 5.86931472424151875016756922329, 7.51773080171603975996014291575, 9.390691972747526801245265580536, 10.66730152363417441123708123421, 11.38026040487639080475052708788, 12.56274054701654781931107198693, 13.35326816699018380033904999717, 14.50700982969809282108683898068