| L(s) = 1 | + (10.8 − 18.7i)2-s + (−13.5 − 23.3i)3-s + (−170. − 295. i)4-s + (−258. + 447. i)5-s − 584.·6-s − 4.61e3·8-s + (−364.5 + 631. i)9-s + (5.59e3 + 9.69e3i)10-s + (−331. − 573. i)11-s + (−4.60e3 + 7.97e3i)12-s + 7.85e3·13-s + 1.39e4·15-s + (−2.81e4 + 4.88e4i)16-s + (−5.81e3 − 1.00e4i)17-s + (7.89e3 + 1.36e4i)18-s + (−7.49e3 + 1.29e4i)19-s + ⋯ |
| L(s) = 1 | + (0.957 − 1.65i)2-s + (−0.288 − 0.499i)3-s + (−1.33 − 2.30i)4-s + (−0.924 + 1.60i)5-s − 1.10·6-s − 3.18·8-s + (−0.166 + 0.288i)9-s + (1.77 + 3.06i)10-s + (−0.0750 − 0.129i)11-s + (−0.769 + 1.33i)12-s + 0.991·13-s + 1.06·15-s + (−1.72 + 2.97i)16-s + (−0.287 − 0.497i)17-s + (0.319 + 0.552i)18-s + (−0.250 + 0.434i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.719398561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719398561\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.5 + 23.3i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (-10.8 + 18.7i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (258. - 447. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (331. + 573. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 7.85e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (5.81e3 + 1.00e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (7.49e3 - 1.29e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.30e4 + 2.26e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.08e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-9.78e4 - 1.69e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.63e5 - 2.82e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 7.91e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.88e5 + 6.72e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-4.64e5 - 8.03e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.14e6 - 1.98e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.21e6 - 2.11e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-8.75e5 - 1.51e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (2.32e5 + 4.01e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (8.66e5 - 1.50e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.91e6 + 6.78e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.49e6T + 8.07e13T^{2} \) |
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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
−11.67671689722388952510482835600, −10.71529492516810748906771989536, −10.36722625762933703150769479944, −8.586174801876376256719918165943, −6.93898311608215495423449152684, −5.93640858755337609278733932742, −4.36496173635525434787727685211, −3.30697009858515842234474833342, −2.48583615380552137908197188421, −0.935336161176410135619719004504,
0.46903833264959969224262404179, 3.71477900098064444757032531323, 4.41852420961095087200008853935, 5.25522681852877883830092379621, 6.28702337978658158386462809208, 7.69168864887245363933314589757, 8.502165227206972893627473662496, 9.201818807685526449523665841558, 11.31824485151436653934694810361, 12.39161174981723757548920807369
