| L(s) = 1 | + (6.69 + 11.5i)2-s + (−25.5 + 44.2i)4-s + (36.7 − 63.5i)5-s + (242. + 419. i)7-s + 1.02e3·8-s + 982.·10-s + (−1.29e3 − 2.24e3i)11-s + (2.98e3 − 5.16e3i)13-s + (−3.23e3 + 5.61e3i)14-s + (1.01e4 + 1.75e4i)16-s + 1.59e4·17-s + 5.14e4·19-s + (1.87e3 + 3.24e3i)20-s + (1.73e4 − 3.00e4i)22-s + (−4.58e4 + 7.94e4i)23-s + ⋯ |
| L(s) = 1 | + (0.591 + 1.02i)2-s + (−0.199 + 0.345i)4-s + (0.131 − 0.227i)5-s + (0.266 + 0.462i)7-s + 0.710·8-s + 0.310·10-s + (−0.293 − 0.509i)11-s + (0.376 − 0.652i)13-s + (−0.315 + 0.546i)14-s + (0.619 + 1.07i)16-s + 0.788·17-s + 1.72·19-s + (0.0524 + 0.0908i)20-s + (0.347 − 0.602i)22-s + (−0.785 + 1.36i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.66980 + 1.86941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.66980 + 1.86941i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-6.69 - 11.5i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-36.7 + 63.5i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-242. - 419. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.29e3 + 2.24e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.98e3 + 5.16e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.58e4 - 7.94e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.26e4 - 7.39e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-6.77e4 + 1.17e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.22e4 + 9.04e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.55e5 + 2.69e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-5.99e5 - 1.03e6i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 7.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-6.74e4 + 1.16e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.39e5 - 7.61e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.52e6 + 2.63e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 4.43e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-2.98e5 - 5.17e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.33e6 + 4.04e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 1.14e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (6.20e6 + 1.07e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.56170342301616087515732149746, −12.27917158770970357187030084481, −11.02481993470055121249566220660, −9.699276385044327744364243691581, −8.222460704241882106616956941563, −7.30459119617944041085878199870, −5.71436850383648735110205173974, −5.26324604177780260385956446028, −3.41354604521947737138826831529, −1.28917262542250503065983188740,
1.14397065115995547204914000360, 2.54902262028986452249406103302, 3.87182317219106932994609749204, 5.06302848225349489217551834086, 6.87561282131634525447258269652, 8.102891055630095513268488842142, 9.880630170102013852562369565748, 10.62885462412874795376304373742, 11.80884997684708726346416123979, 12.46517710184846802042935588861
