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 + (0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-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 + (0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6137468847\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6137468847\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 31 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−13.90200077274478395209904132357, −13.18642218246635535579533856888, −12.17674600461428497912601104499, −10.25696943349034234137116405243, −9.326942655554410349488623302761, −8.536369759493712591970660613850, −7.55840518286483759960699494642, −6.18483581618578829344299054578, −4.55972529911887285331881387887, −3.62548533295285780113509540828,
2.76326840611427196538532991642, 3.23664863338373525323008173673, 5.41607412047591697559067547569, 7.21735457249255791734349745664, 8.302147952705361882193863854075, 9.265677000862623320034465975496, 10.54397566200736196196048454036, 11.32360421178795386961626160752, 12.59387178164138923123330030976, 13.39856948619495352602979483992