L(s) = 1 | + (−1.40 + 2.45i)2-s + (−2.12 − 2.12i)3-s + (−4.04 − 6.90i)4-s + (3.22 − 3.22i)5-s + (8.18 − 2.22i)6-s − 24.6i·7-s + (22.6 − 0.211i)8-s + 8.99i·9-s + (3.37 + 12.4i)10-s + (23.7 − 23.7i)11-s + (−6.06 + 23.2i)12-s + (−18.6 − 18.6i)13-s + (60.3 + 34.6i)14-s − 13.6·15-s + (−31.3 + 55.8i)16-s − 3.55·17-s + ⋯ |
L(s) = 1 | + (−0.497 + 0.867i)2-s + (−0.408 − 0.408i)3-s + (−0.505 − 0.862i)4-s + (0.288 − 0.288i)5-s + (0.557 − 0.151i)6-s − 1.32i·7-s + (0.999 − 0.00936i)8-s + 0.333i·9-s + (0.106 + 0.393i)10-s + (0.651 − 0.651i)11-s + (−0.145 + 0.558i)12-s + (−0.398 − 0.398i)13-s + (1.15 + 0.660i)14-s − 0.235·15-s + (−0.489 + 0.872i)16-s − 0.0506·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.745750 - 0.361995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.745750 - 0.361995i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 2.45i)T \) |
| 3 | \( 1 + (2.12 + 2.12i)T \) |
good | 5 | \( 1 + (-3.22 + 3.22i)T - 125iT^{2} \) |
| 7 | \( 1 + 24.6iT - 343T^{2} \) |
| 11 | \( 1 + (-23.7 + 23.7i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (18.6 + 18.6i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 3.55T + 4.91e3T^{2} \) |
| 19 | \( 1 + (109. + 109. i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 36.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-68.8 - 68.8i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 306.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-92.9 + 92.9i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 385. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-150. + 150. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-451. + 451. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (544. - 544. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-179. - 179. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (283. + 283. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 930. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 701. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 779.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (296. + 296. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 542.T + 9.12e5T^{2} \) |
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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
−15.10603249312558997041237830273, −13.82637869241002386200321070169, −13.11958892267793960152598077743, −11.20515924775881695976794195532, −10.12703612693932687486168986360, −8.693326323759231696278544217702, −7.31655574329489187955159764253, −6.28240020188754860599966046635, −4.62810602663029688101435360330, −0.835139178902662331150660379400,
2.30293058000477669032709386687, 4.40603699311973756928159018010, 6.30901449463204588505969749120, 8.381990700029690853231044459093, 9.539091907954741262706117179311, 10.48618245091943970013782973490, 11.95681705487884624463411589505, 12.38515051054971363071382314052, 14.19119367972926173040669825999, 15.37224535348432962481729498287