L(s) = 1 | + (402. + 123. i)3-s + 1.00e4·5-s + (2.73e4 − 3.50e4i)7-s + (1.46e5 + 9.93e4i)9-s + 2.46e5i·11-s + 1.47e6i·13-s + (4.04e6 + 1.24e6i)15-s + 2.04e6·17-s + 2.01e7i·19-s + (1.53e7 − 1.07e7i)21-s − 3.93e7i·23-s + 5.22e7·25-s + (4.67e7 + 5.80e7i)27-s + 1.09e8i·29-s − 1.28e8i·31-s + ⋯ |
L(s) = 1 | + (0.955 + 0.293i)3-s + 1.43·5-s + (0.614 − 0.788i)7-s + (0.827 + 0.561i)9-s + 0.461i·11-s + 1.09i·13-s + (1.37 + 0.422i)15-s + 0.349·17-s + 1.86i·19-s + (0.819 − 0.573i)21-s − 1.27i·23-s + 1.07·25-s + (0.626 + 0.779i)27-s + 0.993i·29-s − 0.806i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(4.42163 + 1.39443i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.42163 + 1.39443i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-402. - 123. i)T \) |
| 7 | \( 1 + (-2.73e4 + 3.50e4i)T \) |
good | 5 | \( 1 - 1.00e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 2.46e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 1.47e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 2.04e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.01e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 3.93e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.09e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + 1.28e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 1.17e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.13e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.02e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.19e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.82e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 4.66e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.89e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 + 1.10e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.11e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 6.64e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.28e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.28e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.39e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 3.80e10iT - 7.15e21T^{2} \) |
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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
−12.34754351299824165975705679246, −10.57366445236357642487088724192, −9.952686884039124255511652632976, −8.947229368402612900341640385924, −7.74486361998816039989406696920, −6.48508372236400174518034128375, −4.96058559734016632052149588277, −3.79508343506813964315189783508, −2.12736971222976744872466880140, −1.47908987878889205406167039549,
1.06542505921763116718099559006, 2.20028336763190460790600144124, 3.06891273265571114377966969048, 5.07211129537223996666169184591, 6.08349094543359749929421013705, 7.54486557338420978672726938296, 8.763001365748711268138614183033, 9.453480275868370786883280211602, 10.63953199798435706687810244729, 12.11938121060513333618023971504