| L(s) = 1 | + (−1.02e5 − 5.90e4i)2-s + (1.92e7 − 3.85e7i)3-s + (4.82e9 + 8.35e9i)4-s + (7.05e10 + 4.07e10i)5-s + (−4.24e12 + 2.80e12i)6-s + (−3.32e13 + 2.35e11i)7-s + (0.187 − 6.32e14i)8-s + (−1.11e15 − 1.48e15i)9-s + (−4.81e15 − 8.33e15i)10-s + (−1.31e15 + 7.59e14i)11-s + (4.14e17 − 2.51e16i)12-s − 2.46e17·13-s + (3.41e18 + 1.93e18i)14-s + (2.92e18 − 1.93e18i)15-s + (−1.65e19 + 2.87e19i)16-s + (−7.67e19 + 4.42e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.56 − 0.900i)2-s + (0.446 − 0.894i)3-s + (1.12 + 1.94i)4-s + (0.462 + 0.267i)5-s + (−1.50 + 0.993i)6-s + (−0.999 + 0.00709i)7-s − 2.24i·8-s + (−0.600 − 0.799i)9-s + (−0.481 − 0.833i)10-s + (−0.0286 + 0.0165i)11-s + (2.24 − 0.135i)12-s − 0.370·13-s + (1.56 + 0.889i)14-s + (0.445 − 0.294i)15-s + (−0.899 + 1.55i)16-s + (−1.57 + 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.2862583648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2862583648\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.92e7 + 3.85e7i)T \) |
| 7 | \( 1 + (3.32e13 - 2.35e11i)T \) |
| good | 2 | \( 1 + (1.02e5 + 5.90e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-7.05e10 - 4.07e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (1.31e15 - 7.59e14i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 2.46e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (7.67e19 - 4.42e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-7.94e19 + 1.37e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (-6.90e20 - 3.98e20i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 1.45e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (2.06e23 + 3.57e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-4.46e23 + 7.74e23i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 + 9.55e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.25e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (9.02e26 + 5.21e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-2.73e27 + 1.57e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (2.59e28 - 1.49e28i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-4.79e27 + 8.30e27i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-2.27e28 - 3.93e28i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 + 6.68e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-9.75e28 - 1.69e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (1.63e30 - 2.83e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 - 6.04e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (8.39e30 + 4.84e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 - 9.92e31T + 3.77e63T^{2} \) |
| 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
−11.51802977711680544104398305084, −10.25952495267588239561018550243, −9.301026354363844588319011898752, −8.435794791203059561597687000638, −7.17172635043817984601547360864, −6.32147983658626745144983420108, −3.58825300339239614868112651658, −2.44831862894874906552400580510, −1.92574366566378341092285980875, −0.52761937368530253054408057744,
0.15773645074232459555204753422, 1.70962073931463760617335150522, 3.01671992692036284505801260121, 4.87051518547369153097571950860, 6.12168833340710626274133316625, 7.23912029025996342599012097614, 8.580111457778654320174867720351, 9.411015860098939315931080077883, 9.949521046582383477982427422023, 11.16146706446283868766350023800
