L(s) = 1 | + (0.495 + 0.285i)3-s + (−0.355 − 0.205i)5-s − 0.565·7-s + (−1.33 − 2.31i)9-s + 0.623i·11-s + (−3.87 + 2.23i)13-s + (−0.117 − 0.203i)15-s + (−1.03 + 1.79i)17-s + (1.77 − 3.98i)19-s + (−0.280 − 0.161i)21-s + (−2.39 − 4.15i)23-s + (−2.41 − 4.18i)25-s − 3.24i·27-s + (−2.28 + 1.31i)29-s − 5.12·31-s + ⋯ |
L(s) = 1 | + (0.285 + 0.165i)3-s + (−0.159 − 0.0918i)5-s − 0.213·7-s + (−0.445 − 0.771i)9-s + 0.187i·11-s + (−1.07 + 0.620i)13-s + (−0.0303 − 0.0525i)15-s + (−0.251 + 0.434i)17-s + (0.407 − 0.913i)19-s + (−0.0611 − 0.0352i)21-s + (−0.500 − 0.866i)23-s + (−0.483 − 0.836i)25-s − 0.624i·27-s + (−0.424 + 0.245i)29-s − 0.920·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4583801867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4583801867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.77 + 3.98i)T \) |
good | 3 | \( 1 + (-0.495 - 0.285i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.355 + 0.205i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 0.565T + 7T^{2} \) |
| 11 | \( 1 - 0.623iT - 11T^{2} \) |
| 13 | \( 1 + (3.87 - 2.23i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.03 - 1.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.39 + 4.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.28 - 1.31i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 5.19iT - 37T^{2} \) |
| 41 | \( 1 + (-1.11 + 1.93i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.626 - 0.361i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.18 + 7.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.5 - 6.07i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.71 - 4.45i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0512 - 0.0295i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.447 + 0.258i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.32 + 7.49i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.86 + 4.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.50 + 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (4.25 + 7.37i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.33 - 9.24i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.390990061998500433831225362750, −8.712283432838530819782415964538, −7.81793694147073874012800145645, −6.85928341399823269588792141659, −6.19206926167663397755738654385, −4.98897294721528827336119354142, −4.16580105189817192240046783854, −3.13111895732558673730147901043, −2.07611692657607695660986941761, −0.17292206473704570440020272112,
1.82382325901516943993473226159, 2.90353462986698466671899579692, 3.82108243467324725908231702028, 5.17678244765481087839906982680, 5.67069878533802582527923105997, 6.96078768367280843799664325069, 7.76764868483808728474890220468, 8.153593807739948822809615250359, 9.489223153904656869170590317621, 9.782970343168855513525505062477