L(s) = 1 | + (0.980 − 0.195i)2-s + (−0.555 + 0.831i)3-s + (0.923 − 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)6-s + (0.831 − 0.555i)8-s + (−0.382 − 0.923i)9-s − i·10-s + (−0.195 + 0.980i)12-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)16-s + (−0.275 + 0.275i)17-s + (−0.555 − 0.831i)18-s + (1.63 − 0.324i)19-s + (−0.195 − 0.980i)20-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)2-s + (−0.555 + 0.831i)3-s + (0.923 − 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.382 + 0.923i)6-s + (0.831 − 0.555i)8-s + (−0.382 − 0.923i)9-s − i·10-s + (−0.195 + 0.980i)12-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)16-s + (−0.275 + 0.275i)17-s + (−0.555 − 0.831i)18-s + (1.63 − 0.324i)19-s + (−0.195 − 0.980i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.584109396\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.584109396\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.980 + 0.195i)T \) |
| 3 | \( 1 + (0.555 - 0.831i)T \) |
| 5 | \( 1 + (-0.195 + 0.980i)T \) |
good | 7 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (0.275 - 0.275i)T - iT^{2} \) |
| 19 | \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \) |
| 23 | \( 1 + (1.81 - 0.750i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 1.84iT - T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 47 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 53 | \( 1 + (-0.636 + 0.425i)T + (0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 61 | \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 + (-0.750 + 0.149i)T + (0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.16179523301199860090588282058, −9.690387194754962397615339653356, −8.681441668440558485578745960320, −7.53642790843092155090626947921, −6.36308521521590856214055484497, −5.55036868790810303983910466025, −4.97397720529938445589070069306, −4.10504873800469584017851196517, −3.18051071851468865698752679812, −1.49029632519898002964560383913,
1.92591494565642177665614889968, 2.85660281815965038394540389152, 4.04585850220632815545604551623, 5.29634803121018430880244147662, 6.06010652132714985362275715006, 6.63325208923333282535491499829, 7.59080745424292643347159925306, 7.995621721885735127845697301288, 9.707949212320549580282385516651, 10.52995078823927985166977091974