| L(s) = 1 | + (5.24 + 29.7i)2-s + (1.06e3 − 388. i)4-s + (4.78e3 + 4.01e3i)5-s + (6.96e4 + 2.53e4i)7-s + (4.80e4 + 8.32e4i)8-s + (−9.43e4 + 1.63e5i)10-s + (−4.26e5 + 3.58e5i)11-s + (−1.95e5 + 1.10e6i)13-s + (−3.88e5 + 2.20e6i)14-s + (−4.45e5 + 3.74e5i)16-s + (−3.68e6 + 6.39e6i)17-s + (−2.32e6 − 4.03e6i)19-s + (6.65e6 + 2.42e6i)20-s + (−1.28e7 − 1.08e7i)22-s + (2.35e7 − 8.57e6i)23-s + ⋯ |
| L(s) = 1 | + (0.115 + 0.657i)2-s + (0.520 − 0.189i)4-s + (0.684 + 0.574i)5-s + (1.56 + 0.570i)7-s + (0.518 + 0.898i)8-s + (−0.298 + 0.516i)10-s + (−0.799 + 0.670i)11-s + (−0.145 + 0.827i)13-s + (−0.193 + 1.09i)14-s + (−0.106 + 0.0891i)16-s + (−0.630 + 1.09i)17-s + (−0.215 − 0.373i)19-s + (0.465 + 0.169i)20-s + (−0.533 − 0.447i)22-s + (0.763 − 0.277i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.46894 + 3.21220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46894 + 3.21220i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-5.24 - 29.7i)T + (-1.92e3 + 700. i)T^{2} \) |
| 5 | \( 1 + (-4.78e3 - 4.01e3i)T + (8.47e6 + 4.80e7i)T^{2} \) |
| 7 | \( 1 + (-6.96e4 - 2.53e4i)T + (1.51e9 + 1.27e9i)T^{2} \) |
| 11 | \( 1 + (4.26e5 - 3.58e5i)T + (4.95e10 - 2.80e11i)T^{2} \) |
| 13 | \( 1 + (1.95e5 - 1.10e6i)T + (-1.68e12 - 6.12e11i)T^{2} \) |
| 17 | \( 1 + (3.68e6 - 6.39e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.32e6 + 4.03e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.35e7 + 8.57e6i)T + (7.29e14 - 6.12e14i)T^{2} \) |
| 29 | \( 1 + (-6.85e6 - 3.89e7i)T + (-1.14e16 + 4.17e15i)T^{2} \) |
| 31 | \( 1 + (2.27e8 - 8.28e7i)T + (1.94e16 - 1.63e16i)T^{2} \) |
| 37 | \( 1 + (-2.80e8 + 4.86e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-2.80e7 + 1.59e8i)T + (-5.17e17 - 1.88e17i)T^{2} \) |
| 43 | \( 1 + (-8.94e8 + 7.50e8i)T + (1.61e17 - 9.15e17i)T^{2} \) |
| 47 | \( 1 + (1.10e9 + 4.00e8i)T + (1.89e18 + 1.58e18i)T^{2} \) |
| 53 | \( 1 + 2.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-3.57e9 - 2.99e9i)T + (5.23e18 + 2.96e19i)T^{2} \) |
| 61 | \( 1 + (3.05e9 + 1.11e9i)T + (3.33e19 + 2.79e19i)T^{2} \) |
| 67 | \( 1 + (-7.87e8 + 4.46e9i)T + (-1.14e20 - 4.17e19i)T^{2} \) |
| 71 | \( 1 + (-9.40e9 + 1.62e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 + (1.32e10 + 2.29e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.09e9 - 6.18e9i)T + (-7.02e20 + 2.55e20i)T^{2} \) |
| 83 | \( 1 + (-3.89e9 - 2.21e10i)T + (-1.21e21 + 4.40e20i)T^{2} \) |
| 89 | \( 1 + (-3.78e10 - 6.55e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (1.05e11 - 8.83e10i)T + (1.24e21 - 7.04e21i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.49160245249148819631139696899, −11.10177507709166163028706068589, −10.65055148682783566245803799792, −8.956713638358759906916371701790, −7.77302537981358653501179292487, −6.76347242628632069070104544972, −5.60334394899008636943248315617, −4.65691734348798033590413073539, −2.26853184134238843254993086061, −1.83200078097175139238140256460,
0.77498341176351220407154138085, 1.73647412422609776402895414116, 2.91707569314817991212353387123, 4.55278376869431705775698143178, 5.58755655809972679900392101532, 7.34094961872639049077877042744, 8.234320219464160973793908594004, 9.739606454755629197969892486727, 10.97395517803018433178929950936, 11.37095651962809249325329766052
