L(s) = 1 | + (0.610 + 1.87i)2-s + (0.734 + 0.533i)3-s + (−2.34 + 1.70i)4-s + (−0.309 + 0.951i)5-s + (−0.554 + 1.70i)6-s + (−3.03 − 2.20i)8-s + (−0.0542 − 0.166i)9-s − 1.97·10-s + (0.951 − 0.309i)11-s − 2.63·12-s + (0.0966 + 0.297i)13-s + (−0.734 + 0.533i)15-s + (1.39 − 4.29i)16-s + (0.280 − 0.203i)18-s + (0.809 + 0.587i)19-s + (−0.896 − 2.76i)20-s + ⋯ |
L(s) = 1 | + (0.610 + 1.87i)2-s + (0.734 + 0.533i)3-s + (−2.34 + 1.70i)4-s + (−0.309 + 0.951i)5-s + (−0.554 + 1.70i)6-s + (−3.03 − 2.20i)8-s + (−0.0542 − 0.166i)9-s − 1.97·10-s + (0.951 − 0.309i)11-s − 2.63·12-s + (0.0966 + 0.297i)13-s + (−0.734 + 0.533i)15-s + (1.39 − 4.29i)16-s + (0.280 − 0.203i)18-s + (0.809 + 0.587i)19-s + (−0.896 − 2.76i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.397860285\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397860285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.951 + 0.309i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-0.610 - 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + 1.78T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.18410220958735659796557848751, −9.383233599391827635413444663170, −8.710058734583657644840814652336, −8.006401351946517162751969180726, −7.11391577164205411844590975456, −6.49720338638258072560304943364, −5.72360139728395911520147458220, −4.50383815104354791961105028466, −3.65532688693470747825538496140, −3.19288137685339594014827402394,
1.14336232502216317532529679802, 2.05735447972272226059266677709, 3.16672099958379765359420029217, 4.02474067614517818426775362394, 4.91970990933690054419143466871, 5.70803081711080039224429845215, 7.30839054750173424610339610375, 8.518283763896794384860061779569, 8.948098206980913755152264686262, 9.653167189948861518133845868303