L(s) = 1 | + (−2.71 − 0.379i)2-s + (0.249 − 0.0899i)3-s + (5.32 + 1.51i)4-s + (−2.08 + 2.18i)5-s + (−0.711 + 0.150i)6-s + (−0.129 + 0.0400i)7-s + (−8.87 − 3.91i)8-s + (−2.25 + 1.87i)9-s + (6.49 − 5.14i)10-s + (0.0853 + 0.735i)11-s + (1.46 − 0.101i)12-s + (−2.02 − 5.23i)13-s + (0.366 − 0.0597i)14-s + (−0.323 + 0.731i)15-s + (13.2 + 8.19i)16-s + (3.41 − 2.31i)17-s + ⋯ |
L(s) = 1 | + (−1.92 − 0.268i)2-s + (0.143 − 0.0519i)3-s + (2.66 + 0.757i)4-s + (−0.932 + 0.976i)5-s + (−0.290 + 0.0612i)6-s + (−0.0488 + 0.0151i)7-s + (−3.13 − 1.38i)8-s + (−0.751 + 0.623i)9-s + (2.05 − 1.62i)10-s + (0.0257 + 0.221i)11-s + (0.422 − 0.0293i)12-s + (−0.562 − 1.45i)13-s + (0.0979 − 0.0159i)14-s + (−0.0834 + 0.188i)15-s + (3.30 + 2.04i)16-s + (0.827 − 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0240505 - 0.0830803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0240505 - 0.0830803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-3.41 + 2.31i)T \) |
good | 2 | \( 1 + (2.71 + 0.379i)T + (1.92 + 0.547i)T^{2} \) |
| 3 | \( 1 + (-0.249 + 0.0899i)T + (2.30 - 1.91i)T^{2} \) |
| 5 | \( 1 + (2.08 - 2.18i)T + (-0.230 - 4.99i)T^{2} \) |
| 7 | \( 1 + (0.129 - 0.0400i)T + (5.77 - 3.95i)T^{2} \) |
| 11 | \( 1 + (-0.0853 - 0.735i)T + (-10.7 + 2.51i)T^{2} \) |
| 13 | \( 1 + (2.02 + 5.23i)T + (-9.60 + 8.75i)T^{2} \) |
| 19 | \( 1 + (0.150 + 1.08i)T + (-18.2 + 5.19i)T^{2} \) |
| 23 | \( 1 + (1.35 + 4.37i)T + (-18.9 + 12.9i)T^{2} \) |
| 29 | \( 1 + (5.38 + 4.26i)T + (6.63 + 28.2i)T^{2} \) |
| 31 | \( 1 + (8.60 + 0.198i)T + (30.9 + 1.43i)T^{2} \) |
| 37 | \( 1 + (-4.66 + 4.06i)T + (5.11 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.63 + 1.70i)T + (26.1 + 31.5i)T^{2} \) |
| 43 | \( 1 + (-0.0937 - 0.0220i)T + (38.4 + 19.1i)T^{2} \) |
| 47 | \( 1 + (5.29 - 0.490i)T + (46.1 - 8.63i)T^{2} \) |
| 53 | \( 1 + (-3.42 - 4.11i)T + (-9.73 + 52.0i)T^{2} \) |
| 59 | \( 1 + (-6.04 - 4.14i)T + (21.3 + 55.0i)T^{2} \) |
| 61 | \( 1 + (0.0669 + 0.317i)T + (-55.8 + 24.6i)T^{2} \) |
| 67 | \( 1 + (7.57 - 10.0i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-6.70 + 12.7i)T + (-40.1 - 58.5i)T^{2} \) |
| 73 | \( 1 + (3.37 - 2.42i)T + (23.1 - 69.2i)T^{2} \) |
| 79 | \( 1 + (-0.903 + 3.47i)T + (-69.0 - 38.4i)T^{2} \) |
| 83 | \( 1 + (14.5 - 0.672i)T + (82.6 - 7.65i)T^{2} \) |
| 89 | \( 1 + (0.166 - 0.430i)T + (-65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (8.21 + 4.32i)T + (54.8 + 80.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.09241727116649619197805088200, −10.48039232028440428985446776033, −9.602287833777195673743130590391, −8.433852653796863309354131502614, −7.63129302011210313142709272733, −7.28078106072979022212427671132, −5.79897703058492824650408907328, −3.30069774193985795535476760658, −2.45565483106485841479492133395, −0.11247997722320052674372378720,
1.61258602013516481472177705108, 3.56047666794643314415498003240, 5.51840139476854053978004396314, 6.75742646025671565045801544471, 7.74975081387417699648506717442, 8.491098854013019332631538598313, 9.194540561331585168417190505265, 9.863871208995538533908948882861, 11.34904394577793489782097136033, 11.64824143538440054982604406811