L(s) = 1 | + (2.70 − 0.832i)2-s + (−2.12 + 2.12i)3-s + (6.61 − 4.50i)4-s + (11.7 + 11.7i)5-s + (−3.96 + 7.50i)6-s − 12.5i·7-s + (14.1 − 17.6i)8-s − 8.99i·9-s + (41.6 + 22.0i)10-s + (−17.0 − 17.0i)11-s + (−4.47 + 23.5i)12-s + (−49.2 + 49.2i)13-s + (−10.4 − 33.9i)14-s − 50.0·15-s + (23.4 − 59.5i)16-s − 51.8·17-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)2-s + (−0.408 + 0.408i)3-s + (0.826 − 0.562i)4-s + (1.05 + 1.05i)5-s + (−0.269 + 0.510i)6-s − 0.679i·7-s + (0.624 − 0.781i)8-s − 0.333i·9-s + (1.31 + 0.697i)10-s + (−0.466 − 0.466i)11-s + (−0.107 + 0.567i)12-s + (−1.05 + 1.05i)13-s + (−0.199 − 0.648i)14-s − 0.861·15-s + (0.366 − 0.930i)16-s − 0.739·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0172i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.19841 + 0.0189501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19841 + 0.0189501i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.70 + 0.832i)T \) |
| 3 | \( 1 + (2.12 - 2.12i)T \) |
good | 5 | \( 1 + (-11.7 - 11.7i)T + 125iT^{2} \) |
| 7 | \( 1 + 12.5iT - 343T^{2} \) |
| 11 | \( 1 + (17.0 + 17.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (49.2 - 49.2i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 51.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + (11.6 - 11.6i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 74.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-211. + 211. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 326.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-110. - 110. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 348. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-205. - 205. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (225. + 225. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-285. - 285. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-286. + 286. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (627. - 627. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 274. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 298. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 175.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-125. + 125. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 900. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 5.27T + 9.12e5T^{2} \) |
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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
−14.76038270887027542932032213821, −14.10433693684811219754800957929, −13.08044051419644036879867996524, −11.51652142295962159391936624561, −10.57905470852113818186109600532, −9.758990590998877241403553410467, −7.02461419545144466168223969857, −6.03515835304549882214921734738, −4.43853898119637296293849655442, −2.51486336640243487312565473750,
2.23130572978299042511475581909, 5.03137298737696355824324681843, 5.68921650712228447723601379311, 7.34144625178942130654434304871, 8.959043456908851106351359500755, 10.62348198208644885577170445231, 12.40476530488787865276845690961, 12.66989269973618224791061456444, 13.79270304476331236588938189802, 15.12743930603740872930803863562