L(s) = 1 | + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 − 1.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (0.866 + 1.5i)23-s + 25-s − 31-s + (0.499 + 0.866i)36-s + (0.866 + 1.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)43-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 − 1.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (0.866 + 1.5i)23-s + 25-s − 31-s + (0.499 + 0.866i)36-s + (0.866 + 1.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9081482446\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9081482446\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.73T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 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
−9.388283705937069717362030820821, −8.591495608856959468633959486115, −7.87190168069418471209624532358, −7.25092327667428624919529815275, −6.52276719572584919219797994821, −5.24593808881362731225251243419, −4.36435858605497184682018966515, −3.57966518409483424019074569902, −3.11049624973495091304319654380, −1.14651985467179431511887067888,
0.863139815421126946163196074024, 2.20580174082125034035189673830, 3.69114515041586001311617957503, 4.30714991163580986441291342295, 5.19125041171068482694003410426, 5.96584377366271131378868977925, 6.94043380080173136519879875401, 7.75380354711148796049262955677, 8.603037814439052974861915891150, 8.975996713433514968821387944659