L(s) = 1 | + (0.866 + 0.5i)2-s + i·3-s + (0.499 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s − 9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.499i)12-s + (−0.499 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)18-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)21-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + i·3-s + (0.499 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s − 9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.499i)12-s + (−0.499 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)18-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.802962675\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802962675\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−10.20379687934314895488901056916, −9.301910723768656859634613596490, −8.684641072697958856311945394329, −7.42479362567249759098869403724, −6.62136047080126541631488667744, −5.72377583140326229317873768971, −5.23860739767220038446927803754, −4.06571804842304884912483226843, −3.38330758534954897823108681622, −2.31246311293130482964913751944,
1.33749376944039342095287133960, 2.52909892528752814293594857122, 3.03076502324193322716916496028, 4.58871052608067503367866511407, 5.63688792482180903277787507903, 6.28634862325814414317861584422, 6.68223063811269235170266728578, 7.893868956944686925528334457881, 9.093261003901970956352492922937, 9.643402446033453395162874570008