L(s) = 1 | + 3.58·2-s + (0.133 + 15.5i)3-s − 19.1·4-s + 11.8i·5-s + (0.480 + 55.9i)6-s + 150. i·7-s − 183.·8-s + (−242. + 4.17i)9-s + 42.6i·10-s + (352. − 191. i)11-s + (−2.56 − 297. i)12-s + 778. i·13-s + 541. i·14-s + (−185. + 1.59i)15-s − 47.1·16-s + 1.25e3·17-s + ⋯ |
L(s) = 1 | + 0.634·2-s + (0.00859 + 0.999i)3-s − 0.597·4-s + 0.212i·5-s + (0.00545 + 0.634i)6-s + 1.16i·7-s − 1.01·8-s + (−0.999 + 0.0171i)9-s + 0.134i·10-s + (0.878 − 0.477i)11-s + (−0.00513 − 0.597i)12-s + 1.27i·13-s + 0.738i·14-s + (−0.212 + 0.00182i)15-s − 0.0460·16-s + 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.786381 + 1.33523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.786381 + 1.33523i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.133 - 15.5i)T \) |
| 11 | \( 1 + (-352. + 191. i)T \) |
good | 2 | \( 1 - 3.58T + 32T^{2} \) |
| 5 | \( 1 - 11.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 150. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 778. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.21e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 880. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.12e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.39e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.29e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.72e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.10e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 5.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.41e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 9.93e4T + 8.58e9T^{2} \) |
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
−15.86344447432515370667338385924, −14.64615522257171952844736039514, −14.10862233589288455629614118862, −12.33954678408143323784139513844, −11.31538767511600109603705700156, −9.434420591874846042548239166354, −8.812643472837650067409725686787, −6.07889565597513368577601080630, −4.77002155527179426894864912161, −3.25960849837752797013653804291,
0.841618810941280350611269483588, 3.62450022703965031050779871180, 5.47693798597470759334405542154, 7.13221231607963175690306113095, 8.531632501272147968920210245166, 10.27039048652998048748959630616, 12.15086329126730888674533748452, 12.87709192204423397831630777483, 13.99609418752264290236294403721, 14.70835301571275326264406910870