| L(s) = 1 | + (14.4 + 81.8i)2-s + (−4.56e3 + 1.66e3i)4-s + (−352. − 295. i)5-s + (1.36e4 + 4.97e3i)7-s + (−1.16e5 − 2.02e5i)8-s + (1.90e4 − 3.30e4i)10-s + (4.35e5 − 3.65e5i)11-s + (1.38e5 − 7.87e5i)13-s + (−2.09e5 + 1.18e6i)14-s + (7.23e6 − 6.07e6i)16-s + (−4.12e5 + 7.15e5i)17-s + (8.89e6 + 1.53e7i)19-s + (2.09e6 + 7.63e5i)20-s + (3.62e7 + 3.03e7i)22-s + (3.44e7 − 1.25e7i)23-s + ⋯ |
| L(s) = 1 | + (0.318 + 1.80i)2-s + (−2.22 + 0.811i)4-s + (−0.0503 − 0.0422i)5-s + (0.307 + 0.111i)7-s + (−1.25 − 2.18i)8-s + (0.0603 − 0.104i)10-s + (0.815 − 0.684i)11-s + (0.103 − 0.588i)13-s + (−0.104 + 0.590i)14-s + (1.72 − 1.44i)16-s + (−0.0705 + 0.122i)17-s + (0.823 + 1.42i)19-s + (0.146 + 0.0533i)20-s + (1.49 + 1.25i)22-s + (1.11 − 0.405i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.686611 + 2.23788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686611 + 2.23788i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-14.4 - 81.8i)T + (-1.92e3 + 700. i)T^{2} \) |
| 5 | \( 1 + (352. + 295. i)T + (8.47e6 + 4.80e7i)T^{2} \) |
| 7 | \( 1 + (-1.36e4 - 4.97e3i)T + (1.51e9 + 1.27e9i)T^{2} \) |
| 11 | \( 1 + (-4.35e5 + 3.65e5i)T + (4.95e10 - 2.80e11i)T^{2} \) |
| 13 | \( 1 + (-1.38e5 + 7.87e5i)T + (-1.68e12 - 6.12e11i)T^{2} \) |
| 17 | \( 1 + (4.12e5 - 7.15e5i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.89e6 - 1.53e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.44e7 + 1.25e7i)T + (7.29e14 - 6.12e14i)T^{2} \) |
| 29 | \( 1 + (-5.30e6 - 3.01e7i)T + (-1.14e16 + 4.17e15i)T^{2} \) |
| 31 | \( 1 + (-1.90e8 + 6.94e7i)T + (1.94e16 - 1.63e16i)T^{2} \) |
| 37 | \( 1 + (2.38e8 - 4.13e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (2.48e8 - 1.40e9i)T + (-5.17e17 - 1.88e17i)T^{2} \) |
| 43 | \( 1 + (-1.10e9 + 9.30e8i)T + (1.61e17 - 9.15e17i)T^{2} \) |
| 47 | \( 1 + (7.21e8 + 2.62e8i)T + (1.89e18 + 1.58e18i)T^{2} \) |
| 53 | \( 1 - 4.89e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + (4.15e9 + 3.48e9i)T + (5.23e18 + 2.96e19i)T^{2} \) |
| 61 | \( 1 + (-1.07e10 - 3.90e9i)T + (3.33e19 + 2.79e19i)T^{2} \) |
| 67 | \( 1 + (-1.31e9 + 7.43e9i)T + (-1.14e20 - 4.17e19i)T^{2} \) |
| 71 | \( 1 + (8.45e9 - 1.46e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 + (-1.24e10 - 2.15e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (4.77e9 + 2.70e10i)T + (-7.02e20 + 2.55e20i)T^{2} \) |
| 83 | \( 1 + (4.25e9 + 2.41e10i)T + (-1.21e21 + 4.40e20i)T^{2} \) |
| 89 | \( 1 + (1.26e10 + 2.18e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (6.21e10 - 5.21e10i)T + (1.24e21 - 7.04e21i)T^{2} \) |
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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
−12.95373476332879552102674355589, −11.72337303684231938750195330618, −9.963507095872712904083908691409, −8.613528562813180066727003884961, −7.992024214737219477985843484747, −6.67837699662039485297546692536, −5.80620092720616268334534689977, −4.68902781173700788793212501521, −3.41876198637213819599873251660, −0.892350382154489360912124532620,
0.76265102478238239490733211028, 1.75293798111546803396488187807, 2.99821170605681468563766883169, 4.19531675785254751806115101377, 5.14089297830367256350112679345, 7.09011645297686060235723844119, 8.989822897992701662674111622330, 9.579717855295980568496154614769, 10.93871792514778887995649188152, 11.55272444081518934786853897873
