L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (−0.965 + 0.258i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (−0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.965 + 0.258i)21-s − 22-s + (0.866 + 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (−0.965 + 0.258i)5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (−0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.965 + 0.258i)21-s − 22-s + (0.866 + 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1863788100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1863788100\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
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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
−10.57534138549700273824701254905, −9.343521569521577140530785161779, −8.437954089328496844360530336402, −7.74362301488384298923381877507, −6.65367754488528289439482793107, −6.07238930210090469809128350241, −5.16864395241114334128894798503, −4.09185815997014302375133492562, −3.25472734284793241518151277164, −2.24510670294923050317986348610,
0.12023736662908658976329524380, 2.88285712957823136508452782717, 3.95161560093229464119157194336, 4.56317365350444066851361557689, 5.00543379792759014038151019018, 6.16948503291127696767014115328, 6.99311931819981400882675664581, 7.87959902345260126107891311017, 9.041941323856647032222566249603, 9.735951490967885051771759096840