L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.41 + 2.44i)5-s − 0.999·8-s + (1.41 + 2.44i)10-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + (−3.53 + 6.12i)19-s + 2.82·20-s − 1.99·22-s + (−2 + 3.46i)23-s + (−1.49 − 2.59i)25-s − 2·29-s + (4.24 + 7.34i)31-s + (0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.632 + 1.09i)5-s − 0.353·8-s + (0.447 + 0.774i)10-s + (−0.301 − 0.522i)11-s + (−0.125 + 0.216i)16-s + (0.171 + 0.297i)17-s + (−0.811 + 1.40i)19-s + 0.632·20-s − 0.426·22-s + (−0.417 + 0.722i)23-s + (−0.299 − 0.519i)25-s − 0.371·29-s + (0.762 + 1.31i)31-s + (0.0883 + 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.680678 + 0.632955i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.680678 + 0.632955i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.53 - 6.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-4.24 - 7.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 2.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (0.707 + 1.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−10.46347079319739816379525388086, −9.896320536950231223842801760806, −8.551455837573997147368176077014, −7.899509704379604570220817865534, −6.79053802981871482084737415493, −6.02625984748882278786600641477, −4.90423397285373384879936245087, −3.63029029345049699346774820208, −3.19002201008767535183516570283, −1.75523630226212979574070154714,
0.38739379190127039408092108108, 2.37615008901555407300485293327, 3.92203131165310775730625886618, 4.64576506204875424110696668340, 5.35845812057103295600371167702, 6.54419972848753534803873742700, 7.39182482009486717181455059033, 8.259510015734469393054919209140, 8.841512120694845600793720768709, 9.725588064188577853672003316182