| L(s) = 1 | + (−8.50e4 − 4.90e4i)2-s + (−1.46e6 + 4.30e7i)3-s + (2.67e9 + 4.62e9i)4-s + (9.94e10 + 5.74e10i)5-s + (2.23e12 − 3.58e12i)6-s + (−2.61e13 + 2.05e13i)7-s − 1.03e14i·8-s + (−1.84e15 − 1.26e14i)9-s + (−5.63e15 − 9.76e15i)10-s + (3.10e16 − 1.79e16i)11-s + (−2.03e17 + 1.08e17i)12-s + 3.29e17·13-s + (3.23e18 − 4.59e17i)14-s + (−2.61e18 + 4.19e18i)15-s + (6.41e18 − 1.11e19i)16-s + (2.87e19 − 1.65e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.749i)2-s + (−0.0340 + 0.999i)3-s + (0.622 + 1.07i)4-s + (0.651 + 0.376i)5-s + (0.792 − 1.27i)6-s + (−0.786 + 0.617i)7-s − 0.366i·8-s + (−0.997 − 0.0679i)9-s + (−0.563 − 0.976i)10-s + (0.676 − 0.390i)11-s + (−1.09 + 0.585i)12-s + 0.495·13-s + (1.48 − 0.211i)14-s + (−0.398 + 0.638i)15-s + (0.347 − 0.602i)16-s + (0.590 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.08351824829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08351824829\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.46e6 - 4.30e7i)T \) |
| 7 | \( 1 + (2.61e13 - 2.05e13i)T \) |
| good | 2 | \( 1 + (8.50e4 + 4.90e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-9.94e10 - 5.74e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-3.10e16 + 1.79e16i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 - 3.29e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-2.87e19 + 1.65e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-1.00e20 + 1.73e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (-6.94e21 - 4.01e21i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 7.09e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (1.65e23 + 2.86e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (6.45e24 - 1.11e25i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 9.66e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.55e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (2.94e26 + 1.70e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (2.59e27 - 1.49e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-3.96e27 + 2.28e27i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-5.30e27 + 9.18e27i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (1.42e29 + 2.47e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 - 3.66e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-2.00e29 - 3.47e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-3.48e29 + 6.04e29i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 - 6.05e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-5.62e30 - 3.24e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 + 1.17e32T + 3.77e63T^{2} \) |
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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
−11.00678485575578425202542510006, −9.786286584942634978031313891776, −9.418656128352616466948573964176, −8.357283310044019081812583543128, −6.48128430083590269293424508368, −5.27290600011309545632831738453, −3.36552324111477192122261271250, −2.70701792203364956625572754176, −1.29027456765805945752849440906, −0.03067471615103210919836344789,
1.03334058825747992253595537729, 1.65778519186022204978539997748, 3.48060094894927942732605313164, 5.64985444928216099269590212065, 6.63542196835405071171650043435, 7.35415892871181621984137575813, 8.614747592592812547398570801268, 9.476147260105164591918430944345, 10.63067895629204616609230365704, 12.39987802914299382802624779648
