L(s) = 1 | + (0.5 + 0.866i)3-s + 2.21·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (2.45 + 4.24i)11-s + (−2.45 + 2.64i)13-s + (1.10 + 1.92i)15-s + (−0.879 + 1.52i)17-s + (−1.71 + 2.97i)19-s − 0.999·21-s + (0.269 + 0.467i)23-s − 0.0749·25-s − 0.999·27-s + (−2.25 − 3.90i)29-s − 6.38·31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + 0.992·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.739 + 1.28i)11-s + (−0.679 + 0.733i)13-s + (0.286 + 0.496i)15-s + (−0.213 + 0.369i)17-s + (−0.394 + 0.683i)19-s − 0.218·21-s + (0.0562 + 0.0974i)23-s − 0.0149·25-s − 0.192·27-s + (−0.419 − 0.725i)29-s − 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.884418522\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.884418522\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (2.45 - 2.64i)T \) |
good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 + (-2.45 - 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.879 - 1.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.71 - 2.97i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.269 - 0.467i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.25 + 3.90i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 + (1.81 + 3.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.802 - 1.38i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.65 + 2.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.40T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + (-1.36 + 2.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.205 - 0.356i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.30 - 7.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.33 - 2.30i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 + (-6.81 - 11.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.860 - 1.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.408554915265380002347811506575, −8.895558284325829763332159693333, −7.75521337558666856583960933568, −6.95445882485407801009512956479, −6.14220044387317991426200761450, −5.37740505044630422978285508392, −4.42062856680790985107571939272, −3.72721538109103911062829037469, −2.24890758688749863473323652443, −1.85774355585534361348535877286,
0.58543700786187193424330378076, 1.81361204731706377015817327608, 2.83784716917900128840489606167, 3.64743938266397983987861339002, 4.93798519811100184339856917447, 5.79184139642194911162106495115, 6.42338198178067339021657180909, 7.20463085117684974985052526455, 8.002540795012224084613167393287, 9.064088827039748251904089292644