L(s) = 1 | + (2.19 + 3.79i)5-s + (−0.5 + 0.866i)7-s + (2.69 − 4.66i)11-s + (1.27 + 2.20i)13-s + 2.58·17-s + 6.72·19-s + (0.400 + 0.693i)23-s + (−7.09 + 12.2i)25-s + (1.87 − 3.24i)29-s + (1.69 + 2.93i)31-s − 4.38·35-s + 4.38·37-s + (−3.19 − 5.53i)41-s + (−0.381 + 0.661i)43-s + (4.13 − 7.16i)47-s + ⋯ |
L(s) = 1 | + (0.979 + 1.69i)5-s + (−0.188 + 0.327i)7-s + (0.811 − 1.40i)11-s + (0.352 + 0.610i)13-s + 0.625·17-s + 1.54·19-s + (0.0834 + 0.144i)23-s + (−1.41 + 2.45i)25-s + (0.347 − 0.601i)29-s + (0.304 + 0.527i)31-s − 0.740·35-s + 0.720·37-s + (−0.499 − 0.864i)41-s + (−0.0581 + 0.100i)43-s + (0.603 − 1.04i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.582246371\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.582246371\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.19 - 3.79i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.69 + 4.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 2.20i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 + (-0.400 - 0.693i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 3.24i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 2.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 + (3.19 + 5.53i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.381 - 0.661i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 + (2.78 + 4.82i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.14 + 7.17i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.946 + 1.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 + (6.64 - 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.86 - 3.22i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 + (1.48 - 2.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.033178068697149124609330862746, −8.079418641004746843305459692597, −7.09357978942519994596227277818, −6.58318129907186358387373421126, −5.88225470357169157586100050011, −5.39336536835169195185313272003, −3.73742864471056764724299899600, −3.24169903761344562155933634829, −2.41874910870923766428352748612, −1.22917870345209038998300422514,
1.00561266282661451801774813100, 1.51082273483431850333379143014, 2.82596076244788986177645948345, 4.12078682392186730616660447835, 4.73383258220406374972579069592, 5.49878311007369358150457725761, 6.11809112037735926748129888789, 7.16857463480351103806418860236, 7.889332430323087578459293682609, 8.748105372237664210307822028420