L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−1 − i)13-s + (0.965 − 0.258i)14-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)19-s − 0.999i·20-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−1 − i)13-s + (0.965 − 0.258i)14-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)19-s − 0.999i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5594478332\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5594478332\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
| 41 | \( 1 + iT \) |
good | 3 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.811404883985172018493024071428, −8.873510024292664391647033234711, −7.959652056200856276920188754452, −7.33381311248500230038842921724, −6.28273542905788947492743258069, −5.99360730750762572113802779078, −5.01679698314042166956293143654, −3.13653674849270993982662471093, −1.88651350847189259923259994460, −0.65460700553976964454035647515,
2.09912607309863675250599161934, 2.79990947383315099633901214668, 4.13558583923746475716291324653, 4.95946552049353758020768421571, 6.39915769173234334043218242052, 6.85661240686749591606183219748, 7.896758428616271583599478098406, 9.329397267825249841931657802016, 9.534613674452790166954385904040, 10.03402061480628518162239198459