L(s) = 1 | + (−2.24 + 1.29i)2-s + (−1.61 + 0.613i)3-s + (2.36 − 4.09i)4-s − 1.25·5-s + (2.84 − 3.47i)6-s + 7.07i·8-s + (2.24 − 1.98i)9-s + (2.81 − 1.62i)10-s − 0.616i·11-s + (−1.31 + 8.08i)12-s + (1.06 − 0.613i)13-s + (2.02 − 0.769i)15-s + (−4.44 − 7.69i)16-s + (2.21 + 3.83i)17-s + (−2.46 + 7.37i)18-s + (1.64 + 0.950i)19-s + ⋯ |
L(s) = 1 | + (−1.58 + 0.916i)2-s + (−0.935 + 0.354i)3-s + (1.18 − 2.04i)4-s − 0.560·5-s + (1.16 − 1.42i)6-s + 2.50i·8-s + (0.748 − 0.662i)9-s + (0.889 − 0.513i)10-s − 0.185i·11-s + (−0.379 + 2.33i)12-s + (0.294 − 0.170i)13-s + (0.523 − 0.198i)15-s + (−1.11 − 1.92i)16-s + (0.537 + 0.930i)17-s + (−0.581 + 1.73i)18-s + (0.377 + 0.218i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00906659 - 0.201443i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00906659 - 0.201443i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.61 - 0.613i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 1.25T + 5T^{2} \) |
| 11 | \( 1 + 0.616iT - 11T^{2} \) |
| 13 | \( 1 + (-1.06 + 0.613i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 0.950i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.74iT - 23T^{2} \) |
| 29 | \( 1 + (5.07 + 2.93i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.14 - 1.24i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.09 + 3.63i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.24 - 3.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.80 + 6.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.67 + 1.54i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.78 - 3.08i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (12.5 - 7.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (9.95 - 5.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.01 - 3.49i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.36 - 7.56i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.811 + 1.40i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.76 - 5.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.31696205633876209596171802009, −10.38028268456175124991263406578, −9.824234653166443382233659557028, −8.836731571696342939770405875693, −7.87085009527993645091290822273, −7.18812936349087493513463158077, −6.05871429186193692185912777188, −5.49116285662021901800228951975, −3.85608518146072073216043505410, −1.33997237281932266238124819032,
0.26731319228402513152509842397, 1.67472170982464487970497575422, 3.18282999310278406524642284277, 4.68410809654455693457415362966, 6.29317141258522219214012453467, 7.39185030474792412096152423851, 7.86970204060411209372612202282, 9.048667728499112774261927236816, 9.889137334054963169756906582653, 10.71609318426416310204194477371