L(s) = 1 | + (−0.173 + 0.984i)2-s + (1.52 + 1.28i)3-s + (−0.939 − 0.342i)4-s + (−1.52 + 1.28i)6-s + (−1.97 − 3.41i)7-s + (0.5 − 0.866i)8-s + (0.167 + 0.951i)9-s + (2.04 − 3.54i)11-s + (−0.995 − 1.72i)12-s + (1.91 − 1.60i)13-s + (3.70 − 1.34i)14-s + (0.766 + 0.642i)16-s + (−0.884 + 5.01i)17-s − 0.966·18-s + (−1.12 − 4.21i)19-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.880 + 0.739i)3-s + (−0.469 − 0.171i)4-s + (−0.622 + 0.522i)6-s + (−0.745 − 1.29i)7-s + (0.176 − 0.306i)8-s + (0.0559 + 0.317i)9-s + (0.616 − 1.06i)11-s + (−0.287 − 0.497i)12-s + (0.531 − 0.446i)13-s + (0.990 − 0.360i)14-s + (0.191 + 0.160i)16-s + (−0.214 + 1.21i)17-s − 0.227·18-s + (−0.257 − 0.966i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71332 + 0.0599466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71332 + 0.0599466i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.12 + 4.21i)T \) |
good | 3 | \( 1 + (-1.52 - 1.28i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (1.97 + 3.41i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.04 + 3.54i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.91 + 1.60i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.884 - 5.01i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-3.39 - 1.23i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.962 + 5.45i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.07 + 3.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 + (-8.57 - 7.19i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.70 - 1.34i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.680 - 3.85i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (5.31 + 1.93i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 7.74i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (10.6 + 3.87i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.172 + 0.977i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.51 - 0.916i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 2.67i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (2.85 + 2.39i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.87 - 4.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.98 - 8.37i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.982 - 5.57i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.675953003908501798779659906699, −9.303380269481228107890475506584, −8.347605240367667350232536505510, −7.74058886458310527429159443067, −6.50432851786445856040763168349, −6.06616442899210051734010998246, −4.44089983499391253475589724337, −3.83365333572938995832755788795, −3.03926795824862761917785714078, −0.805684059894585978261269044777,
1.57669863578990644664142557540, 2.46745472952267365224256462582, 3.25526036554611081094280271117, 4.53053943278150998333783566477, 5.73763826447586835909499272795, 6.83913811906153297097158332769, 7.55075781145683806533178453787, 8.811557268666472706460665530000, 8.995422672793299318362122519560, 9.778263383553956882418215940028