L(s) = 1 | + i·3-s + (−0.866 + 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (−0.866 − 0.5i)21-s + (−0.866 − 0.5i)25-s − i·27-s − 0.999i·28-s + (1.36 − 0.366i)31-s + (0.866 − 0.5i)36-s + (0.5 − 0.133i)37-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.866 + 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (−0.866 − 0.5i)21-s + (−0.866 − 0.5i)25-s − i·27-s − 0.999i·28-s + (1.36 − 0.366i)31-s + (0.866 − 0.5i)36-s + (0.5 − 0.133i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6086649003\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6086649003\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + iT - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
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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
−12.22157640284510323689059769178, −11.60352512859213456000844825590, −10.12765993759472145699502311956, −9.650958419204735814008440258436, −8.609382485427544086735652332058, −7.980561807734962318510237680934, −6.04903113434626166223002293018, −5.25547027501461647727713418139, −3.93353428475640900203245746300, −3.06952599328811000904381042252,
1.16752905340420434536210547571, 3.29030769500172323723293504840, 4.71738145121298426168457696069, 6.00608123910532792126270461327, 6.90373742155447837391133364102, 7.972053878432931158513995716845, 9.033668792989128623306379253608, 9.861941906027809375403217963769, 11.04152999738188505229706027754, 11.89451732075340779749753212166