L(s) = 1 | + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (0.642 + 0.766i)12-s + (1.58 + 0.424i)13-s + (0.173 + 0.984i)16-s + (−0.642 + 0.766i)18-s + (−0.866 + 0.5i)23-s + (−0.939 + 0.342i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 1.34i)26-s + (−0.866 − 0.5i)27-s + ⋯ |
L(s) = 1 | + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (0.642 + 0.766i)12-s + (1.58 + 0.424i)13-s + (0.173 + 0.984i)16-s + (−0.642 + 0.766i)18-s + (−0.866 + 0.5i)23-s + (−0.939 + 0.342i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 1.34i)26-s + (−0.866 − 0.5i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7341006346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7341006346\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 3 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-1.58 - 0.424i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.515 + 1.92i)T + (-0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (-0.642 - 1.11i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 1.28iT - T^{2} \) |
| 73 | \( 1 - 1.53iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−8.747166653123590568092426059213, −8.059873326480219841095755730893, −7.34094073367821942922870710302, −6.58662513075588384286457942994, −5.80897545005345762631150389525, −5.66145649017699047505802708828, −4.24724949300625033954846303522, −3.99536681908487270651279924024, −2.00151341934609961157236571258, −0.861637822864739925262310365618,
0.885702225483991079135124852995, 1.87719430107657306337371474436, 3.24265916019451836356391656791, 3.97778488682397755181620402242, 4.69575024660133219856958211174, 5.74310074651202867055692940346, 6.21779829994739139412514945250, 7.40872901200970374651820410112, 8.034573740776755519197876621127, 8.951595134149846155823964962994