L(s) = 1 | + (1.15 − 1.99i)2-s + (−1.65 − 2.86i)4-s + (−0.5 − 0.866i)5-s + (1.30 − 2.25i)7-s − 2.99·8-s − 2.30·10-s + (−2.30 + 3.98i)11-s + (−3.30 − 5.72i)13-s + (−2.99 − 5.19i)14-s + (−0.151 + 0.262i)16-s + 1.60·17-s + 3.60·19-s + (−1.65 + 2.86i)20-s + (5.30 + 9.18i)22-s + (1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (0.814 − 1.41i)2-s + (−0.825 − 1.43i)4-s + (−0.223 − 0.387i)5-s + (0.492 − 0.852i)7-s − 1.06·8-s − 0.728·10-s + (−0.694 + 1.20i)11-s + (−0.916 − 1.58i)13-s + (−0.801 − 1.38i)14-s + (−0.0378 + 0.0655i)16-s + 0.389·17-s + 0.827·19-s + (−0.369 + 0.639i)20-s + (1.13 + 1.95i)22-s + (0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.329841 - 1.87062i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.329841 - 1.87062i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.15 + 1.99i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.30 + 2.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.30 - 3.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.30 + 5.72i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.697 - 1.20i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 4.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.30 - 3.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.302 - 0.524i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.60 + 7.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + (-0.697 - 1.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.10 + 3.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.394 - 0.683i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + (-5.80 + 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.87013995750866462450384045352, −10.21998453546656884779765175810, −9.612584528340665716806938834248, −7.921296801756586812359346870499, −7.33781844690131969673872466953, −5.25274025072583409314678167052, −4.90691562500739980656929593506, −3.68591498923956038649469511012, −2.56383472050128163536341837559, −1.04785178700235456643805987958,
2.62236508502945595992006505586, 4.08005808562670108268439947050, 5.14345104924548616574015001239, 5.88505688721990551254759631656, 6.87567547946709116715105814583, 7.75786219646703209756476414367, 8.520110439025083202191459802408, 9.550920167499712036313850780606, 11.01479820020278378828356887099, 11.81134681969740011434754026278