L(s) = 1 | + (2.70 + 1.12i)3-s + (0.151 + 0.366i)5-s + (3.06 − 3.06i)7-s + (3.95 + 3.95i)9-s + (−3.66 + 1.51i)11-s + (0.780 − 1.88i)13-s + 1.16i·15-s + 4.54i·17-s + (0.221 − 0.534i)19-s + (11.7 − 4.86i)21-s + (4.41 + 4.41i)23-s + (3.42 − 3.42i)25-s + (2.91 + 7.03i)27-s + (−4.74 − 1.96i)29-s + 0.0539·31-s + ⋯ |
L(s) = 1 | + (1.56 + 0.647i)3-s + (0.0678 + 0.163i)5-s + (1.15 − 1.15i)7-s + (1.31 + 1.31i)9-s + (−1.10 + 0.457i)11-s + (0.216 − 0.522i)13-s + 0.299i·15-s + 1.10i·17-s + (0.0508 − 0.122i)19-s + (2.56 − 1.06i)21-s + (0.921 + 0.921i)23-s + (0.684 − 0.684i)25-s + (0.560 + 1.35i)27-s + (−0.881 − 0.365i)29-s + 0.00969·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.015769476\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.015769476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-2.70 - 1.12i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.151 - 0.366i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-3.06 + 3.06i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.66 - 1.51i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.780 + 1.88i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + (-0.221 + 0.534i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \) |
| 29 | \( 1 + (4.74 + 1.96i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 0.0539T + 31T^{2} \) |
| 37 | \( 1 + (0.330 + 0.798i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.621 - 0.621i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.06 + 0.857i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 + (10.0 - 4.16i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.97 - 7.17i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (9.72 + 4.02i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-7.53 - 3.12i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (2.99 - 2.99i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.91 + 2.91i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.74iT - 79T^{2} \) |
| 83 | \( 1 + (1.36 - 3.30i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 2.38i)T - 89iT^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−10.09443986847021636383386815864, −9.113312569074007483276417266116, −8.183482407445544528834956593159, −7.82620751933536683746164773151, −7.07001853115789903788525469629, −5.39725491201906973353993657209, −4.49434519717223285005007657538, −3.74727693791982261638563749352, −2.72789263991972311007376459919, −1.59078317005905690169343916762,
1.48175040595203834836084247047, 2.49738505365939639824471338433, 3.11628532011872990353398925362, 4.67371555468132897763592125254, 5.45060143346948635121395251081, 6.76414477778897282909310366144, 7.73720403380310696746073069385, 8.180303748002420538645078998351, 9.070787568274940356412744553716, 9.276457027595863358681814921101