| L(s) = 1 | + (−11.0 − 11.0i)3-s + (−52.2 − 52.2i)5-s + 282.·7-s + 242. i·9-s + (−135. + 135. i)11-s + (−730. + 730. i)13-s + 1.15e3i·15-s + 8.44e3·17-s + (−1.83e3 − 1.83e3i)19-s + (−3.11e3 − 3.11e3i)21-s − 1.44e4·23-s − 1.01e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−2.25e4 + 2.25e4i)29-s + 2.57e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.417 − 0.417i)5-s + 0.823·7-s + 0.333i·9-s + (−0.102 + 0.102i)11-s + (−0.332 + 0.332i)13-s + 0.341i·15-s + 1.71·17-s + (−0.267 − 0.267i)19-s + (−0.336 − 0.336i)21-s − 1.18·23-s − 0.651i·25-s + (0.136 − 0.136i)27-s + (−0.923 + 0.923i)29-s + 0.864i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9408175640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9408175640\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (52.2 + 52.2i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 282.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (135. - 135. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (730. - 730. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 8.44e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (1.83e3 + 1.83e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 1.44e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.25e4 - 2.25e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 2.57e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (2.26e4 + 2.26e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.85e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-1.82e4 + 1.82e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 4.38e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.50e5 + 1.50e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-1.69e5 + 1.69e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (2.12e5 - 2.12e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-6.01e4 - 6.01e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 4.74e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.41e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.15e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (1.79e5 + 1.79e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 5.93e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.72e5T + 8.32e11T^{2} \) |
| 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
−10.63387677601116005629807776928, −9.689194913075976974695986800558, −8.418971814504110626630424600354, −7.82344098253095808590701635062, −6.86950517911093886318218149047, −5.58766796826748378357920780315, −4.84454192533274061922832138368, −3.65931092783679861416868306878, −2.05215006719745819045052026587, −1.01754095880676617428333766000,
0.25339052010525662932739318686, 1.68969327037158147331696098517, 3.20249220289260419828127086799, 4.19440545453059923255028027832, 5.32131402554202300105737460813, 6.11975493814257834706272500060, 7.64077028450691518645102985877, 7.950054074481016040528153104021, 9.422427594357066023800334000980, 10.22021198542466968493880922158
