| L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.437 + 2.48i)3-s + (0.766 − 0.642i)4-s + (−1.88 − 1.58i)5-s + (−0.437 − 2.48i)6-s + (1.39 + 2.41i)7-s + (−0.500 + 0.866i)8-s + (−3.14 − 1.14i)9-s + (2.31 + 0.841i)10-s + (0.839 − 1.45i)11-s + (1.26 + 2.18i)12-s + (1.10 + 6.24i)13-s + (−2.14 − 1.79i)14-s + (4.74 − 3.98i)15-s + (0.173 − 0.984i)16-s + (−4.66 + 1.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.252 + 1.43i)3-s + (0.383 − 0.321i)4-s + (−0.842 − 0.707i)5-s + (−0.178 − 1.01i)6-s + (0.527 + 0.914i)7-s + (−0.176 + 0.306i)8-s + (−1.04 − 0.382i)9-s + (0.730 + 0.266i)10-s + (0.253 − 0.438i)11-s + (0.363 + 0.630i)12-s + (0.305 + 1.73i)13-s + (−0.571 − 0.479i)14-s + (1.22 − 1.02i)15-s + (0.0434 − 0.246i)16-s + (−1.13 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.127312 - 0.398824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.127312 - 0.398824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.437 - 2.48i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.88 + 1.58i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.39 - 2.41i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.839 + 1.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.10 - 6.24i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.66 - 1.69i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (1.91 - 1.60i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.57 + 2.02i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.64 + 6.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.550T + 37T^{2} \) |
| 41 | \( 1 + (0.452 - 2.56i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.19 - 1.84i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.700 + 0.255i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.12 - 0.945i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.66 + 1.69i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.15 + 6.00i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.8 + 3.95i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.35 + 4.49i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.07 - 6.09i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.02 - 5.82i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (7.58 + 13.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.19 - 6.80i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (13.5 - 4.92i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.15989758931765945911268077141, −9.806177336913158070289159542296, −9.027896929772514712817322020075, −8.768402178700234258071736891928, −7.76972088324849313919997216416, −6.41148015479297963173831025269, −5.46986046997882912847624855328, −4.43075302006641954941239756289, −3.90030973896129160877318360382, −1.98032142760905727005646453136,
0.28016338280794088668003523678, 1.55620406314492577526362950473, 2.90257660728378550577283278512, 4.09613535821281787945591135258, 5.67779384112711176368880300238, 7.00016332772369397587463406008, 7.20418527083605094951369598972, 7.925930193070873780647456886168, 8.724240552616270611738427614636, 10.18065977180206813854844847478
