L(s) = 1 | + (−2.17 − 2.06i)3-s + (−3.17 − 3.17i)5-s + 6.03i·7-s + (0.485 + 8.98i)9-s + (13.0 + 13.0i)11-s + (−6.39 − 6.39i)13-s + (0.363 + 13.4i)15-s − 4.39i·17-s + (−3.21 − 3.21i)19-s + (12.4 − 13.1i)21-s + 34.0·23-s − 4.78i·25-s + (17.4 − 20.5i)27-s + (27.9 − 27.9i)29-s + 7.90·31-s + ⋯ |
L(s) = 1 | + (−0.725 − 0.687i)3-s + (−0.635 − 0.635i)5-s + 0.862i·7-s + (0.0539 + 0.998i)9-s + (1.18 + 1.18i)11-s + (−0.491 − 0.491i)13-s + (0.0242 + 0.898i)15-s − 0.258i·17-s + (−0.168 − 0.168i)19-s + (0.593 − 0.626i)21-s + 1.47·23-s − 0.191i·25-s + (0.647 − 0.761i)27-s + (0.964 − 0.964i)29-s + 0.255·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15547 - 0.161704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15547 - 0.161704i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.17 + 2.06i)T \) |
good | 5 | \( 1 + (3.17 + 3.17i)T + 25iT^{2} \) |
| 7 | \( 1 - 6.03iT - 49T^{2} \) |
| 11 | \( 1 + (-13.0 - 13.0i)T + 121iT^{2} \) |
| 13 | \( 1 + (6.39 + 6.39i)T + 169iT^{2} \) |
| 17 | \( 1 + 4.39iT - 289T^{2} \) |
| 19 | \( 1 + (3.21 + 3.21i)T + 361iT^{2} \) |
| 23 | \( 1 - 34.0T + 529T^{2} \) |
| 29 | \( 1 + (-27.9 + 27.9i)T - 841iT^{2} \) |
| 31 | \( 1 - 7.90T + 961T^{2} \) |
| 37 | \( 1 + (20.0 - 20.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 45.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (36.0 - 36.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 5.08iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.7 - 20.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-39.0 - 39.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-49.8 - 49.8i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-44.9 - 44.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 46.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 97.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 40.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-35.5 + 35.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 69.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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
−11.49849758326009878036880311882, −10.20948037544516736154870324024, −9.155514636981195272876803582209, −8.240810219671448356961849007350, −7.22818692088091934051333022928, −6.39807888553071025871453621431, −5.15767979715863848490527975837, −4.40828454444952809729597648300, −2.47614606073918773092927594879, −0.949136701600319310035639421646,
0.831229659229425267144449008025, 3.36471101695236363411232949960, 4.02024693697667400191601125202, 5.23149152812359528799557986786, 6.59968919774170069118563339594, 7.03809553194271480377082292381, 8.529523636990670610129951293959, 9.436603935458656273450839821905, 10.54791243976306203478368430210, 11.10324376507920152792265989792