L(s) = 1 | + (−1.5 + 0.866i)5-s + (2.59 + 1.5i)7-s + (−2.59 + 4.5i)11-s + (−0.5 − 0.866i)13-s − 3.46i·17-s + 6i·19-s + (2.59 + 4.5i)23-s + (−1 + 1.73i)25-s + (7.5 + 4.33i)29-s + (2.59 − 1.5i)31-s − 5.19·35-s − 4·37-s + (−4.5 + 2.59i)41-s + (2.59 + 1.5i)43-s + (2.59 − 4.5i)47-s + ⋯ |
L(s) = 1 | + (−0.670 + 0.387i)5-s + (0.981 + 0.566i)7-s + (−0.783 + 1.35i)11-s + (−0.138 − 0.240i)13-s − 0.840i·17-s + 1.37i·19-s + (0.541 + 0.938i)23-s + (−0.200 + 0.346i)25-s + (1.39 + 0.804i)29-s + (0.466 − 0.269i)31-s − 0.878·35-s − 0.657·37-s + (−0.702 + 0.405i)41-s + (0.396 + 0.228i)43-s + (0.378 − 0.656i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.907896 + 0.761815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.907896 + 0.761815i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 4.5i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (-2.59 - 4.5i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.5 - 4.33i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 1.5i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.59 - 1.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.59 + 4.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.79 - 4.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (12.9 + 7.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.59 + 4.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.51597724793015164133290415858, −10.44972196634010904545536528752, −9.688108547656365290587973775898, −8.402605900652304441406412132627, −7.71048529863858397461564260172, −6.94703234384423004082346582114, −5.38946605234373196976226843857, −4.71045816125836541807855511345, −3.27034398913898410384560579836, −1.89811329018923357890472649700,
0.78667881984835800281641273800, 2.74568514064946745485319457832, 4.21242972291899748900694957756, 4.94293898891516107615446896574, 6.23809981420566085323674454440, 7.43720225186532749497206918951, 8.338096941765726074368802003064, 8.747768950716575098381296446636, 10.37890927582278913951035965066, 10.92519824356104805574338976336