L(s) = 1 | + (364.5 − 210. i)3-s + (3.84e4 + 2.23e4i)7-s + (8.85e4 − 1.53e5i)9-s − 1.45e6i·13-s + (−5.63e6 − 3.25e6i)19-s + (1.87e7 + 5.63e4i)21-s + (2.44e7 + 4.22e7i)25-s − 7.45e7i·27-s + (2.73e8 − 1.57e8i)31-s + (5.96e7 − 1.03e8i)37-s + (−3.05e8 − 5.29e8i)39-s + 2.18e8·43-s + (9.78e8 + 1.71e9i)49-s − 2.73e9·57-s + (−1.07e10 − 6.21e9i)61-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (0.864 + 0.502i)7-s + (0.5 − 0.866i)9-s − 1.08i·13-s + (−0.522 − 0.301i)19-s + (0.999 + 0.00301i)21-s + (0.499 + 0.866i)25-s − 0.999i·27-s + (1.71 − 0.989i)31-s + (0.141 − 0.245i)37-s + (−0.542 − 0.940i)39-s + 0.227·43-s + (0.494 + 0.869i)49-s − 0.602·57-s + (−1.63 − 0.941i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.74648 - 1.83288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.74648 - 1.83288i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-364.5 + 210. i)T \) |
| 7 | \( 1 + (-3.84e4 - 2.23e4i)T \) |
good | 5 | \( 1 + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.45e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (5.63e6 + 3.25e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.22e16T^{2} \) |
| 31 | \( 1 + (-2.73e8 + 1.57e8i)T + (1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-5.96e7 + 1.03e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.18e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (1.07e10 + 6.21e9i)T + (2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.97e9 + 5.15e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.75e10 + 1.59e10i)T + (1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-2.71e10 + 4.70e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.26e11iT - 7.15e21T^{2} \) |
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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
−11.99530083699249016661076669644, −10.75048717822967744100274847660, −9.383785206821784841484280762544, −8.339808346975697969400086992909, −7.59348123397043305801973336736, −6.14140949834141801708416814708, −4.69554157909811064437304811277, −3.14195852561976817387749498775, −2.04907110796179886054655932273, −0.77120180215953608406599383453,
1.29760995943721778137097901463, 2.51878079443057464172103616509, 4.03725895555052673700095112863, 4.82905951358509689272044913458, 6.70654117064118631774547790776, 7.980700800854387752275336793228, 8.802277371994578979957015976349, 10.03842993332988248886229455506, 10.94959930685214184721803372894, 12.19939531135803608820017532140