L(s) = 1 | + (−0.844 − 0.535i)2-s + (−0.317 − 1.23i)3-s + (0.425 + 0.904i)4-s + (−0.535 − 0.844i)5-s + (−0.393 + 1.21i)6-s + (0.394 − 0.996i)7-s + (0.125 − 0.992i)8-s + (−0.547 + 0.301i)9-s + i·10-s + (0.982 − 0.812i)12-s + (−0.866 + 0.629i)14-s + (−0.872 + 0.929i)15-s + (−0.637 + 0.770i)16-s + (0.624 + 0.0392i)18-s + (0.535 − 0.844i)20-s + (−1.35 − 0.171i)21-s + ⋯ |
L(s) = 1 | + (−0.844 − 0.535i)2-s + (−0.317 − 1.23i)3-s + (0.425 + 0.904i)4-s + (−0.535 − 0.844i)5-s + (−0.393 + 1.21i)6-s + (0.394 − 0.996i)7-s + (0.125 − 0.992i)8-s + (−0.547 + 0.301i)9-s + i·10-s + (0.982 − 0.812i)12-s + (−0.866 + 0.629i)14-s + (−0.872 + 0.929i)15-s + (−0.637 + 0.770i)16-s + (0.624 + 0.0392i)18-s + (0.535 − 0.844i)20-s + (−1.35 − 0.171i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5501218427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5501218427\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.844 + 0.535i)T \) |
| 5 | \( 1 + (0.535 + 0.844i)T \) |
| 101 | \( 1 + (-0.992 - 0.125i)T \) |
good | 3 | \( 1 + (0.317 + 1.23i)T + (-0.876 + 0.481i)T^{2} \) |
| 7 | \( 1 + (-0.394 + 0.996i)T + (-0.728 - 0.684i)T^{2} \) |
| 11 | \( 1 + (0.535 - 0.844i)T^{2} \) |
| 13 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 23 | \( 1 + (0.0859 + 1.36i)T + (-0.992 + 0.125i)T^{2} \) |
| 29 | \( 1 + (0.621 + 1.57i)T + (-0.728 + 0.684i)T^{2} \) |
| 31 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 37 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 41 | \( 1 + (-0.700 - 0.227i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.0931 - 0.488i)T + (-0.929 - 0.368i)T^{2} \) |
| 47 | \( 1 + (-0.288 - 1.51i)T + (-0.929 + 0.368i)T^{2} \) |
| 53 | \( 1 + (-0.637 - 0.770i)T^{2} \) |
| 59 | \( 1 + (-0.187 - 0.982i)T^{2} \) |
| 61 | \( 1 + (-0.871 + 0.410i)T + (0.637 - 0.770i)T^{2} \) |
| 67 | \( 1 + (0.0312 - 0.121i)T + (-0.876 - 0.481i)T^{2} \) |
| 71 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 73 | \( 1 + (-0.992 + 0.125i)T^{2} \) |
| 79 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 83 | \( 1 + (1.93 + 0.121i)T + (0.992 + 0.125i)T^{2} \) |
| 89 | \( 1 + (-0.905 + 0.749i)T + (0.187 - 0.982i)T^{2} \) |
| 97 | \( 1 + (0.637 - 0.770i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.736764151558915422797645268988, −7.79307825022081172313816037843, −7.75537821525860619149776707112, −6.80983590625502012366586711990, −6.00410825922531630212482560143, −4.53908102989352179910261358849, −3.93996115991573403941875298272, −2.47936462123950743569410473999, −1.38383927503615489750287469517, −0.58072136751859957899596049292,
1.92568230637588313903153516110, 3.15539378710856752675173957715, 4.15974667947324509550687277678, 5.37690013577986079177375683434, 5.57826676380012916821699384908, 6.83470136819174134161027997371, 7.45912508402388316246553905187, 8.406657390084250751995842199711, 9.053741518542195110395514748628, 9.753052657039160148866231900765