| L(s) = 1 | + (0.991 − 0.130i)2-s + (0.442 + 0.896i)3-s + (0.965 − 0.258i)4-s + (−2.94 + 2.58i)5-s + (0.555 + 0.831i)6-s + (−2.43 + 1.02i)7-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (−2.58 + 2.94i)10-s + (−0.0584 + 0.00382i)11-s + (0.659 + 0.751i)12-s + (−3.92 − 3.92i)13-s + (−2.28 + 1.33i)14-s + (−3.61 − 1.49i)15-s + (0.866 − 0.5i)16-s + (−1.36 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.701 − 0.0922i)2-s + (0.255 + 0.517i)3-s + (0.482 − 0.129i)4-s + (−1.31 + 1.15i)5-s + (0.226 + 0.339i)6-s + (−0.921 + 0.388i)7-s + (0.326 − 0.135i)8-s + (−0.202 + 0.264i)9-s + (−0.816 + 0.931i)10-s + (−0.0176 + 0.00115i)11-s + (0.190 + 0.217i)12-s + (−1.08 − 1.08i)13-s + (−0.609 + 0.357i)14-s + (−0.934 − 0.387i)15-s + (0.216 − 0.125i)16-s + (−0.329 − 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0993687 + 0.864237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0993687 + 0.864237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 3 | \( 1 + (-0.442 - 0.896i)T \) |
| 7 | \( 1 + (2.43 - 1.02i)T \) |
| 17 | \( 1 + (1.36 + 3.89i)T \) |
| good | 5 | \( 1 + (2.94 - 2.58i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (0.0584 - 0.00382i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (3.92 + 3.92i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.607 - 4.61i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.58 + 0.779i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 9.22i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (3.30 - 1.63i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.137 + 2.10i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (2.01 - 10.1i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-1.12 - 2.72i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (3.06 - 11.4i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.05 - 7.89i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-4.06 - 0.535i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.60 - 4.71i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (8.49 + 4.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.77 - 3.19i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (4.96 + 1.68i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 8.01i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 8.36i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (0.739 + 0.198i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (4.91 - 0.976i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.72730245744596185940967743477, −10.26118645113638780666019733771, −9.260521819876874013665445746058, −7.950607974566904281829172218287, −7.32682668493371355297028106191, −6.43408842584421578172588361660, −5.29084077605253460358832208243, −4.20865061861700085507897636253, −3.10101808066482816154836765750, −2.87650177607124742983166384964,
0.32583558890306918623734189714, 2.21592358147835196703823601777, 3.74678341826993688113773040514, 4.26385650935211943025389490848, 5.35551989816302302602788648332, 6.70420651338258323707930934884, 7.24799459649366726984713394159, 8.188283936775295144318051967153, 8.987311848908969064664032551904, 9.937295128484996741951013132924
