L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.766 + 0.642i)6-s + (0.939 + 1.62i)7-s + 0.999·8-s + (0.766 − 0.642i)9-s + (0.173 − 0.984i)12-s + (−0.766 + 1.32i)13-s + (0.939 − 1.62i)14-s + (−0.5 − 0.866i)16-s − 1.87·17-s + (−0.939 − 0.342i)18-s + 19-s + (−1.43 − 1.20i)21-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.766 + 0.642i)6-s + (0.939 + 1.62i)7-s + 0.999·8-s + (0.766 − 0.642i)9-s + (0.173 − 0.984i)12-s + (−0.766 + 1.32i)13-s + (0.939 − 1.62i)14-s + (−0.5 − 0.866i)16-s − 1.87·17-s + (−0.939 − 0.342i)18-s + 19-s + (−1.43 − 1.20i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4849934810\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4849934810\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 23 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.968454737734237479926807202896, −9.094650247307854119345806969220, −8.866390118317072112727827799170, −7.65928128298164776294921211957, −6.71234878205592954463058369361, −5.62359822695586078567176397269, −4.75401072702341736407925833857, −4.17281412466110505596073700109, −2.52283799277445355797043072920, −1.76667867351405197146663164418,
0.54563218880839582343864354639, 1.79658246575294005999879584629, 3.97494465874464303325514597409, 4.91136054886705213173732014430, 5.36888425330129924324546880288, 6.57747278213533847852550794464, 7.29756102652753080263981846072, 7.62248417160936345174091950741, 8.549526784408207323539092101728, 9.748700828335601011902086993390