L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.499i)6-s + i·8-s + (−0.866 − 0.499i)10-s + i·11-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.866 − 0.499i)24-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.499i)6-s + i·8-s + (−0.866 − 0.499i)10-s + i·11-s + (0.866 + 0.5i)13-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.866 − 0.499i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8350514765\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8350514765\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - iT \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−12.59878214374348844857471046594, −11.70170822259938823687102300683, −11.15214454433226423544601240658, −9.384598762562672584106446621976, −8.356505921760740358469166939759, −7.26017384000536900167833545953, −6.05319430970867544832352090710, −4.73452519105335723026706593379, −3.89465344068019022949133309138, −1.88468753944685974228414804407,
3.39534326667322651269387170485, 4.26725185241816332100587755687, 5.51610305641054235868624484660, 6.28222701480822907240696897913, 7.49654642368921679881128986129, 8.896500537691307310573059846531, 10.29690649426814426782369016975, 10.77478702258525130717597088530, 11.71555922436764316537261612012, 13.08864861527252854924438333706