L(s) = 1 | + (0.866 + 0.5i)2-s + (1.86 + 0.5i)3-s + (−0.500 − 0.866i)4-s + (1.5 + 0.866i)5-s + (1.36 + 1.36i)6-s + (−0.866 − 2.5i)7-s − 3i·8-s + (0.633 + 0.366i)9-s + (0.866 + 1.5i)10-s + (4.23 + 1.13i)11-s + (−0.500 − 1.86i)12-s + (−1.73 + 1.73i)13-s + (0.500 − 2.59i)14-s + (2.36 + 2.36i)15-s + (0.500 − 0.866i)16-s + (−1.73 + 6.46i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (1.07 + 0.288i)3-s + (−0.250 − 0.433i)4-s + (0.670 + 0.387i)5-s + (0.557 + 0.557i)6-s + (−0.327 − 0.944i)7-s − 1.06i·8-s + (0.211 + 0.122i)9-s + (0.273 + 0.474i)10-s + (1.27 + 0.341i)11-s + (−0.144 − 0.538i)12-s + (−0.480 + 0.480i)13-s + (0.133 − 0.694i)14-s + (0.610 + 0.610i)15-s + (0.125 − 0.216i)16-s + (−0.420 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29296 + 0.242369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29296 + 0.242369i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.866 + 2.5i)T \) |
| 41 | \( 1 + (-4 - 5i)T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.23 - 1.13i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (1.73 - 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.73 - 6.46i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (5.23 - 1.40i)T + (16.4 - 9.5i)T^{2} \) |
| 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 | \( 1 + (-6.09 + 1.63i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.36 + 0.366i)T + (45.8 + 26.5i)T^{2} \) |
| 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 | \( 1 + (0.633 - 2.36i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.63 + 6.63i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.92 + 2.26i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.76 + 10.3i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 - 0.732T + 83T^{2} \) |
| 89 | \( 1 + (-1.83 - 6.83i)T + (-77.0 + 44.5i)T^{2} \) |
| 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
−12.07730815524410339863574896881, −10.53689110021903210671604485883, −9.843553337107640278516672864842, −9.229733533360545749398438380320, −8.007379402534035487431692728077, −6.57294251021861352931270818101, −6.14329860860340135921247432005, −4.22516877690773971038296850296, −3.83916276343148596910061190062, −1.94491572347120926009242755511,
2.27873613165005906994138549951, 2.97870646947108745985572245694, 4.41378305849073306847147518385, 5.57779236430090422374774983672, 6.86403595923064404910108923358, 8.336134923729881378600474461197, 8.869475138896169607852078543565, 9.493408330377044358819768442134, 11.09784283393501639732312597874, 12.23902073518511083763384998711