| L(s) = 1 | + (−1.03e5 − 5.99e4i)2-s + (−9.97e6 − 4.18e7i)3-s + (5.03e9 + 8.72e9i)4-s + (−1.39e11 − 8.06e10i)5-s + (−1.47e12 + 4.94e12i)6-s + (2.67e13 + 1.97e13i)7-s + (0.125 − 6.92e14i)8-s + (−1.65e15 + 8.35e14i)9-s + (9.66e15 + 1.67e16i)10-s + (2.54e15 − 1.46e15i)11-s + (3.15e17 − 2.97e17i)12-s − 1.02e18·13-s + (−1.59e18 − 3.65e18i)14-s + (−1.98e18 + 6.65e18i)15-s + (−1.98e19 + 3.44e19i)16-s + (1.64e19 − 9.47e18i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 − 0.914i)2-s + (−0.231 − 0.972i)3-s + (1.17 + 2.03i)4-s + (−0.914 − 0.528i)5-s + (−0.522 + 1.75i)6-s + (0.804 + 0.593i)7-s − 2.46i·8-s + (−0.892 + 0.450i)9-s + (0.966 + 1.67i)10-s + (0.0554 − 0.0319i)11-s + (1.70 − 1.61i)12-s − 1.53·13-s + (−0.732 − 1.67i)14-s + (−0.301 + 1.01i)15-s + (−1.07 + 1.86i)16-s + (0.337 − 0.194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.07033721825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07033721825\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (9.97e6 + 4.18e7i)T \) |
| 7 | \( 1 + (-2.67e13 - 1.97e13i)T \) |
| good | 2 | \( 1 + (1.03e5 + 5.99e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (1.39e11 + 8.06e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-2.54e15 + 1.46e15i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 1.02e18T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-1.64e19 + 9.47e18i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (2.00e20 - 3.47e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (-1.36e21 - 7.89e20i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 - 2.17e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-2.63e23 - 4.56e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (9.66e24 - 1.67e25i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 1.22e26iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.75e25T + 1.86e52T^{2} \) |
| 47 | \( 1 + (6.98e26 + 4.03e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (6.47e26 - 3.73e26i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-2.81e28 + 1.62e28i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-2.26e28 + 3.92e28i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-6.60e28 - 1.14e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 + 2.34e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (3.15e29 + 5.45e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (7.60e29 - 1.31e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 - 4.42e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (2.57e30 + 1.48e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 + 1.04e32T + 3.77e63T^{2} \) |
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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
−11.30606531298727104705053676281, −9.902014989665364999233400568868, −8.346584303866445540456433470183, −8.114609981233216815602600049251, −6.93830792109409845099235221850, −4.95041945569415323221441135129, −3.08753497630387489083311036635, −1.95417705485536200911534856849, −1.19780469714474090557036788230, −0.04915578582203025497908818539,
0.54685089892111070326880515786, 2.33692526585829819728695093103, 4.14854467956849354116332246661, 5.34273465013378281810799732835, 6.92210595240091881963151383612, 7.67923215732408398749483543695, 8.780911726293202726757379047636, 9.953051055719053317503882223872, 10.80255711145024258610285370442, 11.64423777364806567652057776563
