L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.923 − 0.382i)3-s + (0.499 + 0.866i)4-s + (−0.608 − 0.793i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s − 1.93i·11-s + (−0.130 − 0.991i)12-s + (−0.5 + 0.866i)16-s + (0.608 − 1.05i)17-s + (0.258 + 0.965i)18-s + (−0.226 + 0.130i)19-s + (0.965 − 1.67i)22-s + (0.382 − 0.923i)24-s + 25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.923 − 0.382i)3-s + (0.499 + 0.866i)4-s + (−0.608 − 0.793i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s − 1.93i·11-s + (−0.130 − 0.991i)12-s + (−0.5 + 0.866i)16-s + (0.608 − 1.05i)17-s + (0.258 + 0.965i)18-s + (−0.226 + 0.130i)19-s + (0.965 − 1.67i)22-s + (0.382 − 0.923i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.614036897\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614036897\) |
\(L(1)\) |
|
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 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.93iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.226 - 0.130i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.793 + 1.37i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2} \) |
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
−8.486549748989926426170999804149, −7.79864848254603335780348340173, −7.05680123394557450449115691713, −6.38959119434462281014770008655, −5.64831966427323050854033196896, −5.28482126709549358436843903890, −4.29381574095543254058237399404, −3.35895511368304457402626432246, −2.49769727647164410105879985559, −0.921493865745403836527414212466,
1.27565428476547551069688612606, 2.23920510060906541986671976859, 3.47904135966012681327620473719, 4.32800178522911004346513613347, 4.80200084240021305645407505183, 5.55410576164901694812545268913, 6.40639010992344142316917841616, 6.93538985520167913746605066334, 7.75341212326892945386487083675, 9.061350979824586307124480097045