L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + 0.999i·10-s + (0.866 − 0.499i)12-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)21-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + 0.999i·10-s + (0.866 − 0.499i)12-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128384359\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128384359\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−9.677663251565390690266378308485, −9.054230090117232234452526822583, −8.435743381208330882298664561413, −7.938697976147735330692125054863, −6.92983442540591483575354288157, −5.60858894865506845331071828690, −5.14505044005394401730894640656, −4.04683879173593199218158197754, −2.37061325829712497862036464563, −1.62813366163584216738510460794,
1.51546696754779595838707432141, 2.29262332592900602287258308256, 3.33412832730430024173773760450, 4.22307055571076778221071183088, 5.94943247201834410264958344523, 6.85112768910182457192654440551, 7.71917664933308835196392695141, 7.950190498533994535869496874178, 9.128081390025512962206391347285, 9.712946577131832680071317606780