L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.965 − 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (−0.258 − 0.965i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.258 + 0.965i)18-s + i·19-s + (1.22 + 1.22i)22-s + (0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.965 − 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (−0.258 − 0.965i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (−0.258 + 0.965i)18-s + i·19-s + (1.22 + 1.22i)22-s + (0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8858299358\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8858299358\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 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
−10.65427956932928628926149291078, −9.786658380750722333654077956878, −8.834226499925666687571611685444, −8.340207736389128160026029759172, −7.62970450650671747037675368682, −6.46893148745807972454904486372, −5.82769030661011163315792003108, −4.29461748777089127943875183278, −3.00787396981305638761693367608, −1.47224940725126572218738898426,
1.91540564081001050958220064208, 2.76206245112357439677472686171, 4.08650844877286557769597642430, 4.84021912317408969478257054777, 7.05094998844366798044334666795, 7.31868632952487026059281293649, 8.553362702842214825505602013615, 9.193505144241107666651545853368, 9.882752347528073464154965589982, 10.58466898194519248950448277548