L(s) = 1 | + (0.717 − 1.21i)2-s + (1.11 + 1.32i)3-s + (−0.971 − 1.74i)4-s + (0.337 − 0.195i)5-s + (2.41 − 0.401i)6-s + (1.39 − 2.24i)7-s + (−2.82 − 0.0704i)8-s + (−0.529 + 2.95i)9-s + (0.00458 − 0.551i)10-s + (−0.748 + 1.29i)11-s + (1.24 − 3.23i)12-s + 3.28·13-s + (−1.73 − 3.31i)14-s + (0.634 + 0.232i)15-s + (−2.11 + 3.39i)16-s + (−1.68 + 2.91i)17-s + ⋯ |
L(s) = 1 | + (0.507 − 0.861i)2-s + (0.641 + 0.766i)3-s + (−0.485 − 0.874i)4-s + (0.151 − 0.0872i)5-s + (0.986 − 0.164i)6-s + (0.527 − 0.849i)7-s + (−0.999 − 0.0249i)8-s + (−0.176 + 0.984i)9-s + (0.00144 − 0.174i)10-s + (−0.225 + 0.390i)11-s + (0.358 − 0.933i)12-s + 0.911·13-s + (−0.464 − 0.885i)14-s + (0.163 + 0.0599i)15-s + (−0.528 + 0.848i)16-s + (−0.407 + 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55418 - 0.660921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55418 - 0.660921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.717 + 1.21i)T \) |
| 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 7 | \( 1 + (-1.39 + 2.24i)T \) |
good | 5 | \( 1 + (-0.337 + 0.195i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.748 - 1.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.56 + 4.43i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.72 - 2.72i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + (3.60 + 2.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.46 + 4.31i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 4.79iT - 43T^{2} \) |
| 47 | \( 1 + (2.51 + 4.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.499 - 0.864i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.36 - 0.785i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.40 - 5.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.239 + 0.414i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 + (-2.54 - 4.41i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.00iT - 97T^{2} \) |
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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
−13.05544679212223144781047519927, −11.26734029236166023118629178524, −10.87267836071858866350301473627, −9.799869400436738809930094962868, −8.929749721887275897312341524218, −7.68779094598316587303878462038, −5.83690754241104142686159390298, −4.48311518166199324488276551382, −3.73653766108167827150187749057, −2.01204065205952192747816406954,
2.43401401642573580471252798073, 3.99175898905261210996230515162, 5.71353656411123395638906381090, 6.44671677562867397671723077230, 7.88756967181435679821969597360, 8.411852968014419067716508023808, 9.427914396327494359059316020086, 11.29284068373055175806778196402, 12.30755213175570381997785737226, 13.06184779044745487196500673411