L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)14-s + (0.866 + 0.499i)15-s + 16-s + (0.866 + 0.499i)18-s + (0.5 + 0.866i)20-s + (0.499 − 0.866i)21-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)14-s + (0.866 + 0.499i)15-s + 16-s + (0.866 + 0.499i)18-s + (0.5 + 0.866i)20-s + (0.499 − 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5450172804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5450172804\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.827142723922235708005386607142, −8.921887978690508670810853649537, −8.579129380909278776780231294553, −7.28920037498478581718301121702, −6.56447244494153824135735209759, −5.78154781719610375319774185790, −4.95673341691327456159294137010, −4.33763849536350140362342013284, −3.24002861636102620612676847763, −0.68719966956534640315122733631,
1.08080326584813666627281231480, 2.68024968985736277918995971606, 3.49482649977898740589664514074, 4.55291394193142573463709777775, 5.53567221924180413816539885178, 6.64994049661565549806628346198, 7.18760602323423438448103720617, 8.157597091426351334397369733704, 9.284536260744595207816879745696, 10.15937901279612542394120872549