| L(s) = 1 | + (1.41 + 0.0577i)2-s + (−1.73 − 0.0359i)3-s + (1.99 + 0.163i)4-s + (−3.11 − 2.69i)5-s + (−2.44 − 0.150i)6-s + (−0.723 − 1.12i)7-s + (2.80 + 0.345i)8-s + (2.99 + 0.124i)9-s + (−4.24 − 3.98i)10-s + (0.835 − 5.80i)11-s + (−3.44 − 0.354i)12-s + (−1.70 − 1.09i)13-s + (−0.957 − 1.63i)14-s + (5.29 + 4.78i)15-s + (3.94 + 0.650i)16-s + (−0.406 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0408i)2-s + (−0.999 − 0.0207i)3-s + (0.996 + 0.0815i)4-s + (−1.39 − 1.20i)5-s + (−0.998 − 0.0615i)6-s + (−0.273 − 0.425i)7-s + (0.992 + 0.122i)8-s + (0.999 + 0.0415i)9-s + (−1.34 − 1.26i)10-s + (0.251 − 1.75i)11-s + (−0.994 − 0.102i)12-s + (−0.471 − 0.303i)13-s + (−0.255 − 0.436i)14-s + (1.36 + 1.23i)15-s + (0.986 + 0.162i)16-s + (−0.0986 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0466 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0466 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.868194 - 0.909722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.868194 - 0.909722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0577i)T \) |
| 3 | \( 1 + (1.73 + 0.0359i)T \) |
| 23 | \( 1 + (3.89 + 2.79i)T \) |
| good | 5 | \( 1 + (3.11 + 2.69i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.723 + 1.12i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.835 + 5.80i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.70 + 1.09i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.406 - 1.38i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 3.61i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.416 - 1.41i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-5.92 + 2.70i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.91 - 2.21i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.20 - 2.77i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.657 - 0.300i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 + (3.83 + 5.96i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.91 - 2.51i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (3.02 + 6.61i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-11.0 + 1.59i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.669 - 4.65i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (5.15 - 1.51i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (3.06 - 4.77i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (3.12 + 3.60i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.814 - 0.372i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.96 + 4.57i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.73376555570313363241432276111, −11.22260519449544776564782752735, −10.13092482005057507301194900806, −8.391999307336710940462856285136, −7.65325503957726606247431897148, −6.35891080432901856088974819594, −5.43769770392457251592156725555, −4.36062945317687008748688756355, −3.60244272443291110362011515546, −0.814262444753987085902705289030,
2.51182783881141224879744573264, 4.02205192242866676487670373075, 4.74841542928816870442884199431, 6.21070034061034022643675614442, 7.15537185020273576440453809869, 7.47856310483334852746304522038, 9.729123962681036414444529532998, 10.62145102710673701340664797132, 11.54128371716881846365074753159, 12.02966510979713495958833009741
