L(s) = 1 | + 1.22·2-s + (−4.91 − 1.67i)3-s − 6.50·4-s + (9.89 − 5.21i)5-s + (−6.01 − 2.04i)6-s + (−1.55 + 18.4i)7-s − 17.7·8-s + (21.4 + 16.4i)9-s + (12.0 − 6.37i)10-s + 54.5i·11-s + (32.0 + 10.8i)12-s − 24.4·13-s + (−1.90 + 22.5i)14-s + (−57.3 + 9.09i)15-s + 30.3·16-s + 36.0i·17-s + ⋯ |
L(s) = 1 | + 0.432·2-s + (−0.946 − 0.322i)3-s − 0.813·4-s + (0.884 − 0.466i)5-s + (−0.409 − 0.139i)6-s + (−0.0841 + 0.996i)7-s − 0.783·8-s + (0.792 + 0.609i)9-s + (0.382 − 0.201i)10-s + 1.49i·11-s + (0.769 + 0.261i)12-s − 0.522·13-s + (−0.0363 + 0.430i)14-s + (−0.987 + 0.156i)15-s + 0.474·16-s + 0.513i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0728 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0728 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.607585 + 0.653558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.607585 + 0.653558i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.91 + 1.67i)T \) |
| 5 | \( 1 + (-9.89 + 5.21i)T \) |
| 7 | \( 1 + (1.55 - 18.4i)T \) |
good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 11 | \( 1 - 54.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 24.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 99.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 97.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 186. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 199. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 514. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 451.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 355.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 656. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 73.7iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 611. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 424.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 814.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 585. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 272.T + 9.12e5T^{2} \) |
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
−13.26260502023974140073843355418, −12.43411384603477498749080201682, −12.07037589541300946061728005710, −10.10512973678741701298914806988, −9.542046022647849384770189353002, −8.069512192374756372162264368695, −6.29755975071203480033634607950, −5.43885298497457419444341364039, −4.48813761260439556300813140551, −1.90796872178350465262848159755,
0.49749988105666134184196493778, 3.44151091645560458361559380250, 4.86901400778772563480556885226, 5.87153765773719271517987999871, 7.00576035140359838995599978101, 8.914405970659879888465468722682, 10.01206213011122021107025774007, 10.78945185996494567999068131898, 11.95416915871345978842509069563, 13.38136387842914802467297791941