L(s) = 1 | + (−0.866 + 1.11i)2-s + (−1.58 − 0.707i)3-s + (−0.500 − 1.93i)4-s + 1.41i·5-s + (2.15 − 1.15i)6-s + (1 − 2.44i)7-s + (2.59 + 1.11i)8-s + (2.00 + 2.23i)9-s + (−1.58 − 1.22i)10-s + 3.46·11-s + (−0.578 + 3.41i)12-s + 3.16·13-s + (1.87 + 3.23i)14-s + (1.00 − 2.23i)15-s + (−3.5 + 1.93i)16-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.790i)2-s + (−0.912 − 0.408i)3-s + (−0.250 − 0.968i)4-s + 0.632i·5-s + (0.881 − 0.471i)6-s + (0.377 − 0.925i)7-s + (0.918 + 0.395i)8-s + (0.666 + 0.745i)9-s + (−0.500 − 0.387i)10-s + 1.04·11-s + (−0.167 + 0.985i)12-s + 0.877·13-s + (0.500 + 0.865i)14-s + (0.258 − 0.577i)15-s + (−0.875 + 0.484i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.702281 + 0.126447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.702281 + 0.126447i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 1.11i)T \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 8.94iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−12.98698935573171699705199334199, −11.44660980732982149032236396219, −10.87132427074034086495142486431, −9.966106980842757381221658165376, −8.585318219331452332265900928007, −7.28726159394183099936430138769, −6.75877462132178894306185104227, −5.66451042194735201129528785452, −4.22733816769738465197811367851, −1.23722799363660427789192439325,
1.38987445863716406825841116413, 3.66417686544298282379807499248, 4.94522912674983041577673598076, 6.21315878066231990403790202327, 7.83221433622793464894810279399, 9.173134829294250687950791214434, 9.489257450418973368004839171692, 11.10354990346480605200890235019, 11.49873031458325967165090874288, 12.39005636359807383043766648779