L(s) = 1 | + (−0.866 + 1.5i)2-s + (−1 − 1.73i)4-s + (0.5 − 0.866i)5-s − i·7-s + 1.73·8-s + (−0.5 + 0.866i)9-s + (0.866 + 1.5i)10-s + (0.5 + 0.866i)11-s + 1.73·13-s + (1.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)18-s − 2·20-s − 1.73·22-s + (−0.499 − 0.866i)25-s + (−1.49 + 2.59i)26-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.5i)2-s + (−1 − 1.73i)4-s + (0.5 − 0.866i)5-s − i·7-s + 1.73·8-s + (−0.5 + 0.866i)9-s + (0.866 + 1.5i)10-s + (0.5 + 0.866i)11-s + 1.73·13-s + (1.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)18-s − 2·20-s − 1.73·22-s + (−0.499 − 0.866i)25-s + (−1.49 + 2.59i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5760973225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5760973225\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−11.46080312419225115368258588896, −10.39159299449733706589013088568, −9.632488800304817831215167313342, −8.673708067663213308904072583051, −8.109973185374966185473690221380, −7.10225986056623281774499995091, −6.16691698685384104176392115118, −5.26642473036905962726009626598, −4.16963397743594160483724965970, −1.42174566134066064097801232296,
1.61210524922490722420013501106, 3.06635690080505824694319942541, 3.56758278003002425810435719087, 5.78000739916433869084460896989, 6.55093440885118146151958984469, 8.376806705253618141933037361807, 8.841004371062147885114769381425, 9.608448759889419473048506859367, 10.64073585774487106238709784502, 11.35170113641093832165645419652