L(s) = 1 | + (0.445 + 0.895i)2-s + (−1.45 − 1.32i)3-s + (−0.602 + 0.798i)4-s + (0.538 − 1.89i)6-s + (−0.982 − 0.183i)8-s + (0.264 + 2.85i)9-s + (0.329 − 0.436i)11-s + (1.93 − 0.361i)12-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)17-s + (−2.43 + 1.50i)18-s + (1.67 − 1.03i)19-s + (0.538 + 0.100i)22-s + (1.18 + 1.56i)24-s + (−0.602 − 0.798i)25-s + ⋯ |
L(s) = 1 | + (0.445 + 0.895i)2-s + (−1.45 − 1.32i)3-s + (−0.602 + 0.798i)4-s + (0.538 − 1.89i)6-s + (−0.982 − 0.183i)8-s + (0.264 + 2.85i)9-s + (0.329 − 0.436i)11-s + (1.93 − 0.361i)12-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)17-s + (−2.43 + 1.50i)18-s + (1.67 − 1.03i)19-s + (0.538 + 0.100i)22-s + (1.18 + 1.56i)24-s + (−0.602 − 0.798i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7528537524\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7528537524\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.445 - 0.895i)T \) |
| 137 | \( 1 + (0.850 + 0.526i)T \) |
good | 3 | \( 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2} \) |
| 5 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 7 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 11 | \( 1 + (-0.329 + 0.436i)T + (-0.273 - 0.961i)T^{2} \) |
| 13 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 17 | \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 1.03i)T + (0.445 - 0.895i)T^{2} \) |
| 23 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 29 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 31 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.47T + T^{2} \) |
| 43 | \( 1 + (-0.465 + 0.288i)T + (0.445 - 0.895i)T^{2} \) |
| 47 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 53 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \) |
| 61 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 67 | \( 1 + (1.83 - 0.710i)T + (0.739 - 0.673i)T^{2} \) |
| 71 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 0.533i)T + (0.739 + 0.673i)T^{2} \) |
| 79 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 83 | \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \) |
| 89 | \( 1 + (0.757 - 1.52i)T + (-0.602 - 0.798i)T^{2} \) |
| 97 | \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \) |
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
−10.11864029280291824356262795313, −8.961821249008484241405147816224, −7.86424804899329801156280825078, −7.35274382092026704829781035410, −6.65754276062146007408100484117, −5.77999858426905841978911898350, −5.37594853503530810763174188744, −4.30595114711303291283656133383, −2.70424124909361677927354410308, −0.900915257119487722157573742750,
1.22498040784660269594954162759, 3.26242005906740536600379169593, 3.98114809083941476858757533245, 4.82191055497729377747646909442, 5.63925807074550287559022130245, 6.09115363618053615708015513047, 7.45709908640402639582061047621, 9.087728912695262912076239763430, 9.676453392838187785029009133857, 10.12264082181509800696600710246