L(s) = 1 | + (0.866 − 0.5i)2-s − i·3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s − 9-s + 0.999i·10-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.499i)14-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s − i·19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s − i·3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s − 9-s + 0.999i·10-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.866 − 0.499i)14-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s − i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8428212999\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8428212999\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−12.86885423872195339372756305172, −12.07904995420139766374280688809, −11.25650667323700744381805010794, −10.22986453644125780396496968362, −8.603510286675660809996359554799, −7.22636910134097705140984556518, −6.88567482910557282915102155226, −5.06485414514885459340030530931, −3.63948583156517302384840049644, −2.54179315234813575415517087046,
3.40231692162563963078878105280, 4.49483680603590198979315620537, 5.43205268266639321517703319445, 6.32975189902255865903931757235, 8.146453088111622490090656980144, 9.261724102180448573909808369698, 9.830151359881541042159863280871, 11.46165193614767954918030784073, 12.20908379768565418980757631090, 13.23814494038459796268873593366