L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (0.499 − 0.866i)12-s + 13-s + (−0.866 + 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (−0.499 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (0.499 − 0.866i)12-s + 13-s + (−0.866 + 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (−0.499 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9861272036\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9861272036\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.739930733848846027100836868817, −8.707485537907556182224276023518, −8.157115406167944104793415862167, −7.76779917214864303131093747684, −6.67020143741414806485178411719, −5.96918583449952107772271012661, −4.98196415723888310553439424804, −3.99683462019422730492580051042, −3.30418694169560511483829351739, −1.95424411317784814303678629580,
0.969793678303853713740495673871, 1.40022008222686113157421918342, 2.93710872304117844276231169433, 3.87175596324403517275366368581, 4.65103177652120886567261198089, 5.56807657324010805308057169073, 7.20499487507028761099019446363, 7.75353739747279958529293730500, 8.168126148931965735662544149300, 9.112721111818720592719437093911