L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + 0.999·10-s + (0.866 − 1.5i)11-s + (−0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + 0.999·10-s + (0.866 − 1.5i)11-s + (−0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378649804\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378649804\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 11 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−10.62611138855508366626772859024, −9.611421201162191241520264294814, −8.834516954960309173679637103018, −7.21166199670469244856740826187, −6.37799371952172503933052054609, −5.83079246285896163769018546423, −5.30615498532723023529307643933, −3.74072631333149780240910048620, −2.67945446327995201365151969259, −1.49873120028330709055817509635,
1.84774347150073181453977262847, 3.69642798530385028376381949398, 4.40195000398156122297160016410, 5.22469127331136095437667209674, 6.14568021332520550600200028443, 6.82308217092216047736537199136, 7.58430099452878429725493001337, 9.184240323326301836362392350024, 9.676022978899620014449995002017, 10.58579487189787782904390392953