| L(s) = 1 | + (−1.32 − 0.505i)2-s + (−0.0284 + 0.179i)3-s + (1.48 + 1.33i)4-s + (1.98 + 1.03i)5-s + (0.128 − 0.223i)6-s + 0.839i·7-s + (−1.29 − 2.51i)8-s + (2.82 + 0.916i)9-s + (−2.09 − 2.36i)10-s + (−1.13 + 2.23i)11-s + (−0.282 + 0.230i)12-s + (−0.0565 − 0.110i)13-s + (0.423 − 1.10i)14-s + (−0.242 + 0.327i)15-s + (0.439 + 3.97i)16-s + (−0.867 − 0.630i)17-s + ⋯ |
| L(s) = 1 | + (−0.934 − 0.357i)2-s + (−0.0164 + 0.103i)3-s + (0.744 + 0.667i)4-s + (0.886 + 0.463i)5-s + (0.0524 − 0.0911i)6-s + 0.317i·7-s + (−0.457 − 0.889i)8-s + (0.940 + 0.305i)9-s + (−0.662 − 0.749i)10-s + (−0.343 + 0.673i)11-s + (−0.0815 + 0.0664i)12-s + (−0.0156 − 0.0307i)13-s + (0.113 − 0.296i)14-s + (−0.0626 + 0.0844i)15-s + (0.109 + 0.993i)16-s + (−0.210 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991701 + 0.363292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991701 + 0.363292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.505i)T \) |
| 5 | \( 1 + (-1.98 - 1.03i)T \) |
| good | 3 | \( 1 + (0.0284 - 0.179i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 - 0.839iT - 7T^{2} \) |
| 11 | \( 1 + (1.13 - 2.23i)T + (-6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (0.0565 + 0.110i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.867 + 0.630i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.56 - 0.248i)T + (18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (3.63 - 1.18i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.00352 - 0.0222i)T + (-27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (-4.71 - 3.42i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.85 + 1.96i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.07 - 1.00i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.41 - 2.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.85 + 4.98i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.13 + 0.812i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (3.69 - 1.88i)T + (34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (-2.92 - 1.48i)T + (35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-5.76 + 0.913i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (9.05 + 12.4i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (12.0 - 3.90i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.394 + 0.286i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (13.4 - 2.13i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-7.89 + 2.56i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.73 + 4.89i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.04987140220325186881260305015, −10.24805110040455556875953794801, −9.780633859803874414244016213064, −8.852829449253523312631417955415, −7.66909734105726668225717109434, −6.90085758982845049457949973655, −5.84599067783495219252022160740, −4.34536868428240637245283094968, −2.72461377309643431788580050915, −1.72420444255660097391647690210,
1.01658835009336988496418961159, 2.40459393313841135615186908415, 4.38366310696083445736271595773, 5.76217915641849739974552903740, 6.45641085724898373054293119523, 7.54534771082593295683676717902, 8.484415338270401662275739912115, 9.370936060844434913140831700127, 10.13043463015257274522058724220, 10.78665555523405180109840804662
