L(s) = 1 | + (−0.569 − 0.392i)2-s + (0.885 − 0.464i)3-s + (−0.539 − 1.42i)4-s + (1.21 − 1.37i)5-s + (−0.686 − 0.0833i)6-s + (1.07 − 4.35i)7-s + (−0.582 + 2.36i)8-s + (0.568 − 0.822i)9-s + (−1.23 + 0.304i)10-s + (−0.960 + 0.663i)11-s + (−1.13 − 1.00i)12-s + (−3.13 − 1.77i)13-s + (−2.32 + 2.05i)14-s + (0.440 − 1.78i)15-s + (−1.01 + 0.900i)16-s + (4.40 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.277i)2-s + (0.511 − 0.268i)3-s + (−0.269 − 0.711i)4-s + (0.545 − 0.615i)5-s + (−0.280 − 0.0340i)6-s + (0.405 − 1.64i)7-s + (−0.206 + 0.836i)8-s + (0.189 − 0.274i)9-s + (−0.390 + 0.0962i)10-s + (−0.289 + 0.199i)11-s + (−0.328 − 0.291i)12-s + (−0.869 − 0.493i)13-s + (−0.620 + 0.549i)14-s + (0.113 − 0.461i)15-s + (−0.254 + 0.225i)16-s + (1.06 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.500751 - 1.23337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.500751 - 1.23337i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.885 + 0.464i)T \) |
| 13 | \( 1 + (3.13 + 1.77i)T \) |
good | 2 | \( 1 + (0.569 + 0.392i)T + (0.709 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.21 + 1.37i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (-1.07 + 4.35i)T + (-6.19 - 3.25i)T^{2} \) |
| 11 | \( 1 + (0.960 - 0.663i)T + (3.90 - 10.2i)T^{2} \) |
| 17 | \( 1 + (-4.40 - 1.08i)T + (15.0 + 7.90i)T^{2} \) |
| 19 | \( 1 - 5.11iT - 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + (0.496 - 0.718i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (5.84 + 0.710i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (-6.15 - 0.747i)T + (35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (4.28 + 8.17i)T + (-23.2 + 33.7i)T^{2} \) |
| 43 | \( 1 + (0.376 + 3.10i)T + (-41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (-5.53 - 2.09i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-1.65 - 0.407i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-6.74 + 7.61i)T + (-7.11 - 58.5i)T^{2} \) |
| 61 | \( 1 + (-3.34 + 0.823i)T + (54.0 - 28.3i)T^{2} \) |
| 67 | \( 1 + (11.9 + 4.52i)T + (50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (-5.37 - 10.2i)T + (-40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (-5.03 + 3.47i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-1.61 + 4.25i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (5.78 - 11.0i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 + (7.71 + 8.70i)T + (-11.6 + 96.2i)T^{2} \) |
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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
−10.14803015784172865115662953272, −10.00355957001201179359182083637, −8.909418564219998997945714656260, −7.88162851572850757513897756971, −7.25646338270911742684603192518, −5.73779521574939350421689825584, −4.92799136140095133625473189224, −3.69178148257321268944442214747, −1.91459233262501541938724887041, −0.898655295303844625041731848718,
2.40009290950412329882690444538, 3.04158919220928715976201355279, 4.65110584220798617473994115006, 5.67027232938130954392681791976, 6.89138759313282439581573233771, 7.79112994554379250687373778952, 8.685506924553838017193218305464, 9.318434314682471783908214267766, 9.974021801643918090108793239027, 11.30360403273326353160506671534