| 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
−9.525365675353469251178233188979, −8.655686048783363265671573121506, −7.933248264347861907343524335759, −7.33809979909948547377294954545, −6.67848713555740294766762851378, −5.24052408171944572091309802022, −4.63839098119612566944703350748, −3.06328765487530502363088019828, −1.69334326259693896225317087175, −0.35897088519317638691418093905,
1.82775399028981729634837854036, 3.15038291946174663036513372505, 4.02035086013791936980148636946, 4.69910961123417354241678952218, 6.43498422911825560495116530011, 7.07156598378051276308184583411, 8.081089303613767369082229019241, 8.783379743073458402409211335532, 9.773582679740721258142806976510, 10.21369111861453985033930721555
