L(s) = 1 | + (0.247 − 0.429i)2-s + (−1.37 + 1.05i)3-s + (0.877 + 1.51i)4-s + (1.84 + 3.19i)5-s + (0.109 + 0.851i)6-s + 1.86·8-s + (0.792 − 2.89i)9-s + 1.83·10-s + (0.446 − 0.772i)11-s + (−2.80 − 1.17i)12-s + (0.598 + 1.03i)13-s + (−5.90 − 2.46i)15-s + (−1.29 + 2.23i)16-s + 0.249·17-s + (−1.04 − 1.05i)18-s − 2.80·19-s + ⋯ |
L(s) = 1 | + (0.175 − 0.303i)2-s + (−0.795 + 0.606i)3-s + (0.438 + 0.759i)4-s + (0.825 + 1.43i)5-s + (0.0447 + 0.347i)6-s + 0.658·8-s + (0.264 − 0.964i)9-s + 0.579·10-s + (0.134 − 0.233i)11-s + (−0.809 − 0.337i)12-s + (0.165 + 0.287i)13-s + (−1.52 − 0.636i)15-s + (−0.323 + 0.559i)16-s + 0.0606·17-s + (−0.246 − 0.249i)18-s − 0.644·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0816 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0816 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01502 + 1.10153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01502 + 1.10153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.37 - 1.05i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.598 - 1.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.249T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.79 + 3.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.88T + 53T^{2} \) |
| 59 | \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.40 - 9.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 1.83T + 73T^{2} \) |
| 79 | \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.16 + 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + (-5.52 + 9.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.33351999271225933462356065823, −10.51959339371228334099079042471, −10.03625931654977398932746899547, −8.790974395333280635065089875301, −7.39269443847993015429546206658, −6.52278905471682625127679768710, −5.92113563595524810073609742450, −4.36226633448583058888473604336, −3.34560794291852927557069697089, −2.21543250087331032586365939137,
1.04989572030583147234040521039, 2.01410289020566243792709939357, 4.58078339347544583296054707708, 5.33920474807743627590497015156, 5.99998303689274576875069346925, 6.88583449384823033540900754065, 8.035808525651127278887563720707, 9.129199072987003165495469736383, 10.10178965160664400426250128499, 10.84280226305239612683995182226