L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 1.86i)3-s + (−0.500 − 0.866i)4-s + (−1.5 − 0.866i)5-s + (1.36 − 1.36i)6-s + (2.5 − 0.866i)7-s + 3i·8-s + (−0.633 − 0.366i)9-s + (0.866 + 1.5i)10-s + (−1.13 + 4.23i)11-s + (1.86 − 0.500i)12-s + (−1.73 − 1.73i)13-s + (−2.59 − 0.500i)14-s + (2.36 − 2.36i)15-s + (0.500 − 0.866i)16-s + (6.46 + 1.73i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 + 1.07i)3-s + (−0.250 − 0.433i)4-s + (−0.670 − 0.387i)5-s + (0.557 − 0.557i)6-s + (0.944 − 0.327i)7-s + 1.06i·8-s + (−0.211 − 0.122i)9-s + (0.273 + 0.474i)10-s + (−0.341 + 1.27i)11-s + (0.538 − 0.144i)12-s + (−0.480 − 0.480i)13-s + (−0.694 − 0.133i)14-s + (0.610 − 0.610i)15-s + (0.125 − 0.216i)16-s + (1.56 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.586261 + 0.362137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.586261 + 0.362137i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.5 + 0.866i)T \) |
| 41 | \( 1 + (-4 + 5i)T \) |
good | 2 | \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.5 - 1.86i)T + (-2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.13 - 4.23i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-6.46 - 1.73i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.40 - 5.23i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.73 - 6.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.73 - 6.73i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 - 5.46iT - 43T^{2} \) |
| 47 | \( 1 + (1.63 + 6.09i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 + 1.36i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.06 + 3.5i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.36 - 0.633i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.63 - 6.63i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.92 - 2.26i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 2.76i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 - 0.732T + 83T^{2} \) |
| 89 | \( 1 + (6.83 - 1.83i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (5.46 - 5.46i)T - 97iT^{2} \) |
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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
−11.81151233532647929085609757097, −10.67225329160758405280215518966, −10.11162570006157599471553525786, −9.578291529219052329323295133800, −8.105951383870364837114297515224, −7.67899981794446296763254117928, −5.44704789319198853249801868324, −4.89562546218098390002880026239, −3.84005659900998156308865243467, −1.55642298247629892690490389631,
0.73395917718888594622106665778, 2.86680015480360249496612369712, 4.44032515007122986701165539456, 5.97889491626366333250614007184, 7.09818640262132342470421212234, 7.83581135101102614985336308868, 8.326900051272817066089826906977, 9.549120531605300005956522106739, 10.88610915345473683383838910983, 11.95434727341129138128628095659