L(s) = 1 | + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (0.190 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s + 0.618·23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯ |
L(s) = 1 | + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (0.190 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s + 0.618·23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.136097591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136097591\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−9.533973482195947095773430384360, −8.565380925853737839181462616412, −7.33913931956396647031770622194, −7.09366521541492215140423604863, −5.78026527202102135929968583138, −5.22468041487568057483296394102, −4.28606516916897942383683105997, −3.24588558521075082405944708887, −2.64410846358512875662186522678, −0.902967577569815416861286500863,
1.51168696704987709529697247973, 2.25098831899809015604863774115, 3.89692965906837451583909724345, 5.01299527421013749686078410533, 5.56919691033676277124904450063, 6.33566622316266720514617648492, 6.94408335931463795429198706644, 7.74093007752102663656858409026, 8.229076582719228173938044452713, 9.602465803014763586778516472643