L(s) = 1 | + (0.831 − 0.555i)2-s + (0.195 − 0.980i)3-s + (0.382 − 0.923i)4-s + (0.555 + 0.831i)5-s + (−0.382 − 0.923i)6-s + (−0.195 − 0.980i)8-s + (−0.923 − 0.382i)9-s + (0.923 + 0.382i)10-s + (−0.831 − 0.555i)12-s + (0.923 − 0.382i)15-s + (−0.707 − 0.707i)16-s + (−0.980 − 0.195i)17-s + (−0.980 + 0.195i)18-s + (0.707 + 1.70i)19-s + (0.980 − 0.195i)20-s + ⋯ |
L(s) = 1 | + (0.831 − 0.555i)2-s + (0.195 − 0.980i)3-s + (0.382 − 0.923i)4-s + (0.555 + 0.831i)5-s + (−0.382 − 0.923i)6-s + (−0.195 − 0.980i)8-s + (−0.923 − 0.382i)9-s + (0.923 + 0.382i)10-s + (−0.831 − 0.555i)12-s + (0.923 − 0.382i)15-s + (−0.707 − 0.707i)16-s + (−0.980 − 0.195i)17-s + (−0.980 + 0.195i)18-s + (0.707 + 1.70i)19-s + (0.980 − 0.195i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.761631761\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761631761\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.831 + 0.555i)T \) |
| 3 | \( 1 + (-0.195 + 0.980i)T \) |
| 5 | \( 1 + (-0.555 - 0.831i)T \) |
| 17 | \( 1 + (0.980 + 0.195i)T \) |
good | 7 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.275 + 1.38i)T + (-0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (1.17 - 1.17i)T - iT^{2} \) |
| 53 | \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
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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
−10.14879397727536827475681843727, −9.309221257557225195325399915041, −8.141887804137639036988008279245, −7.17556533158327626357883409258, −6.30136879068307137114683043568, −5.95228885462337359787527826473, −4.63457475144626477070010893850, −3.33278225076566734948955948851, −2.53754969307520753550051482809, −1.55763646793779086736871781000,
2.26724841138352773518928903509, 3.38979797900242376220086562452, 4.49123439311256932468167175413, 5.02112473268988302766704835739, 5.81872121025408207618246988080, 6.80716004949114209547464658491, 7.942108785139008275143674462914, 8.789983475573752990288133922188, 9.325296166169812706773322033023, 10.26906977502458454910884055961