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 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−11.10324376507920152792265989792, −10.54791243976306203478368430210, −9.436603935458656273450839821905, −8.529523636990670610129951293959, −7.03809553194271480377082292381, −6.59968919774170069118563339594, −5.23149152812359528799557986786, −4.02024693697667400191601125202, −3.36471101695236363411232949960, −0.831229659229425267144449008025,
0.949136701600319310035639421646, 2.47614606073918773092927594879, 4.40828454444952809729597648300, 5.15767979715863848490527975837, 6.39807888553071025871453621431, 7.22818692088091934051333022928, 8.240810219671448356961849007350, 9.155514636981195272876803582209, 10.20948037544516736154870324024, 11.49849758326009878036880311882