L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s − 0.999·21-s + (0.866 + 0.5i)23-s + (0.499 − 0.866i)24-s − 0.999i·27-s − 0.999i·28-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s − 0.999·21-s + (0.866 + 0.5i)23-s + (0.499 − 0.866i)24-s − 0.999i·27-s − 0.999i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5677749731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5677749731\) |
\(L(1)\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.42700848390621985474927384794, −9.394435600847425396542626112748, −8.449296826747297693673260201261, −7.46699946029881942192492962638, −7.19817905569662309929529293930, −5.99723983351032494088585004851, −5.34042349022179951479056514552, −4.30177137695258627828954065354, −2.22188553346566853230913988191, −1.02067260218162582006938144824,
1.29540585027610531385593859983, 2.75969226503359838135478581940, 4.08113616362854792679244335778, 5.02573130791424161470127936589, 6.07712454519151878211262297271, 7.06771007184409688590845993697, 7.974913179066763630309007449262, 8.961920343272576124571207253254, 9.489595480128461231313880652289, 10.57412507204392932604122735996