L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.416 + 1.28i)3-s + (0.309 + 0.951i)4-s + (−1.66 + 1.49i)5-s + (0.416 − 1.28i)6-s + 0.369·7-s + (0.309 − 0.951i)8-s + (0.956 − 0.694i)9-s + (2.22 − 0.228i)10-s + (0.816 + 0.593i)11-s + (−1.09 + 0.792i)12-s + (−3.34 + 2.43i)13-s + (−0.298 − 0.217i)14-s + (−2.60 − 1.51i)15-s + (−0.809 + 0.587i)16-s + (−2.15 + 6.62i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.240 + 0.740i)3-s + (0.154 + 0.475i)4-s + (−0.744 + 0.667i)5-s + (0.170 − 0.523i)6-s + 0.139·7-s + (0.109 − 0.336i)8-s + (0.318 − 0.231i)9-s + (0.703 − 0.0722i)10-s + (0.246 + 0.178i)11-s + (−0.314 + 0.228i)12-s + (−0.927 + 0.673i)13-s + (−0.0798 − 0.0580i)14-s + (−0.673 − 0.390i)15-s + (−0.202 + 0.146i)16-s + (−0.522 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.179150 + 0.635875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.179150 + 0.635875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (1.66 - 1.49i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.416 - 1.28i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 0.369T + 7T^{2} \) |
| 11 | \( 1 + (-0.816 - 0.593i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.34 - 2.43i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.15 - 6.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (0.217 + 0.157i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.572 + 1.76i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.000268 - 0.000825i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.82 + 2.05i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.06 - 6.58i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 + (1.70 + 5.24i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.78 - 5.49i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.0924 - 0.0671i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.78 + 3.47i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (1.80 - 5.54i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.74 - 5.38i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (4.66 + 3.38i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.03 + 6.25i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.56 - 10.9i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.159 - 0.115i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.69 - 8.28i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.21369111861453985033930721555, −9.773582679740721258142806976510, −8.783379743073458402409211335532, −8.081089303613767369082229019241, −7.07156598378051276308184583411, −6.43498422911825560495116530011, −4.69910961123417354241678952218, −4.02035086013791936980148636946, −3.15038291946174663036513372505, −1.82775399028981729634837854036,
0.35897088519317638691418093905, 1.69334326259693896225317087175, 3.06328765487530502363088019828, 4.63839098119612566944703350748, 5.24052408171944572091309802022, 6.67848713555740294766762851378, 7.33809979909948547377294954545, 7.933248264347861907343524335759, 8.655686048783363265671573121506, 9.525365675353469251178233188979