L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.707 + 0.707i)7-s + 0.999i·8-s + (−0.866 − 0.499i)10-s + (1.67 + 0.965i)11-s − 0.517i·13-s + (0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s + 0.999·20-s − 1.93·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.258 + 0.448i)26-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.707 + 0.707i)7-s + 0.999i·8-s + (−0.866 − 0.499i)10-s + (1.67 + 0.965i)11-s − 0.517i·13-s + (0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s + 0.999·20-s − 1.93·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.258 + 0.448i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9240783326\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9240783326\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 0.517iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.93T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−9.216237264856224246400702437177, −8.834663221678083644935300008776, −7.58507527932948461079783563867, −6.84073356592730121839971855289, −6.54688892568354864746964720106, −5.69326254784258689227085664427, −4.79475961534372534270447220791, −3.32971801145566140139655572103, −2.51070452709945652462439586662, −1.38541520389998747984391472629,
0.961966542610762246690355993679, 1.65986085681675450538809245649, 3.30085242671608875807848883241, 3.70533003084745293872709218108, 4.87155005416509833532904590749, 6.06797216158891330711320647042, 6.71608759042470779182542236553, 7.42140134910258185348495023349, 8.522449943698295144731052049956, 9.009408077514181151160149469685