L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·7-s − 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (1.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (1.5 − 0.866i)19-s + 0.999i·20-s + 1.73·22-s + (0.866 + 1.5i)23-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·7-s − 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (1.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (1.5 − 0.866i)19-s + 0.999i·20-s + 1.73·22-s + (0.866 + 1.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.950144512\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.950144512\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−8.924524408387203219212522508618, −7.63985727824363842518199486483, −7.02455051564084810450155568284, −6.41217493346591099073310849898, −5.48385437577559445680035891407, −4.83975519717312877725775588946, −3.84129532783944073800099835614, −3.19006218128439522249964301483, −2.51749112283799657582074771132, −1.12698989008650369483884476709,
1.14176143051824671957532774105, 2.68867170495868369797191142281, 3.62838545457613604099238695839, 4.22287628654929464589595622200, 4.88115834249788289495860604611, 5.70461180186815982924957793717, 6.62935187131587883566591089596, 7.35288217183346809192826387220, 7.76827181456494508840892396497, 8.647326159483015458388452716294