L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.10 + 0.767i)5-s + (0.823 − 1.42i)7-s + 0.999·8-s + (1.71 + 1.43i)10-s + (−2.08 + 1.20i)11-s + (−3.59 − 0.256i)13-s − 1.64·14-s + (−0.5 − 0.866i)16-s + (−0.210 − 0.121i)17-s + (3.82 + 2.20i)19-s + (0.385 − 2.20i)20-s + (2.08 + 1.20i)22-s + (7.46 − 4.31i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.939 + 0.343i)5-s + (0.311 − 0.538i)7-s + 0.353·8-s + (0.542 + 0.453i)10-s + (−0.628 + 0.362i)11-s + (−0.997 − 0.0710i)13-s − 0.439·14-s + (−0.125 − 0.216i)16-s + (−0.0511 − 0.0295i)17-s + (0.877 + 0.506i)19-s + (0.0861 − 0.492i)20-s + (0.444 + 0.256i)22-s + (1.55 − 0.898i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9801004119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9801004119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.10 - 0.767i)T \) |
| 13 | \( 1 + (3.59 + 0.256i)T \) |
good | 7 | \( 1 + (-0.823 + 1.42i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.210 + 0.121i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.82 - 2.20i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.46 + 4.31i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0221 + 0.0383i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 + (-4.47 - 7.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.210 - 0.121i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.82 + 3.36i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + (-8.35 - 4.82i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.937 - 1.62i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.53 - 3.77i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 6.79T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + (-8.69 + 5.02i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.25 + 14.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.992644466402484186504655398273, −8.935928518919712445547249137267, −8.055160959108486513211431534437, −7.42089318536742812640623485305, −6.80285858363259092688853676710, −5.12955783916437104349970716830, −4.49587182145587707848590423644, −3.36422041838138340684672794974, −2.52140495678339852271842149963, −0.870334850968159669680348967219,
0.70408615940391018632584839452, 2.48769828230872838315769937020, 3.73054984427531087639604943651, 5.10554331026410560619962803000, 5.24827093067618312696789833111, 6.69707700084139833716292590172, 7.58356077007524557947730605289, 7.948372826797698136277953452422, 9.042262225715370975134152524010, 9.420699568258027531482783999986