L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.499i)20-s + (0.499 + 0.866i)25-s + (0.866 + 0.499i)26-s + (0.866 + 1.5i)29-s + (0.866 + 0.499i)32-s − 0.999·34-s + (1.5 − 0.866i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.499i)20-s + (0.499 + 0.866i)25-s + (0.866 + 0.499i)26-s + (0.866 + 1.5i)29-s + (0.866 + 0.499i)32-s − 0.999·34-s + (1.5 − 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9538093771\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9538093771\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 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
−9.228902043956974448023595887908, −8.592950710894910851609871975237, −7.60147187138419152512691292028, −7.16911942509595140106013255103, −6.12717119897831650680003560538, −5.68189923823844491408262784235, −4.82577927304522875852214016488, −3.25152875438690647159265353840, −2.37067363576697715237417509971, −1.21645416768467545754152740481,
1.05554800917851117845016537503, 2.13562098277091642969601317697, 2.94357341823626712641425667179, 4.21273960370464163810336898745, 5.04071965813992517362004318794, 6.19450713636957629346343554878, 6.77927983456496355626659899432, 7.82782100681435645854092285493, 8.387844353644418589844820426383, 9.269104641270713047944104754465