| L(s) = 1 | + (−1.34 + 0.445i)2-s + (0.601 + 1.18i)3-s + (1.60 − 1.19i)4-s + (−2.23 − 0.138i)5-s + (−1.33 − 1.31i)6-s + (−0.707 − 0.707i)7-s + (−1.61 + 2.32i)8-s + (0.731 − 1.00i)9-s + (3.05 − 0.808i)10-s + (−1.48 − 2.04i)11-s + (2.37 + 1.17i)12-s + (0.469 + 2.96i)13-s + (1.26 + 0.633i)14-s + (−1.17 − 2.71i)15-s + (1.13 − 3.83i)16-s + (3.39 + 1.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 + 0.315i)2-s + (0.347 + 0.681i)3-s + (0.801 − 0.598i)4-s + (−0.998 − 0.0620i)5-s + (−0.544 − 0.537i)6-s + (−0.267 − 0.267i)7-s + (−0.572 + 0.820i)8-s + (0.243 − 0.335i)9-s + (0.966 − 0.255i)10-s + (−0.448 − 0.617i)11-s + (0.686 + 0.338i)12-s + (0.130 + 0.822i)13-s + (0.337 + 0.169i)14-s + (−0.304 − 0.701i)15-s + (0.284 − 0.958i)16-s + (0.824 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.883901 + 0.0759361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.883901 + 0.0759361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 0.445i)T \) |
| 5 | \( 1 + (2.23 + 0.138i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (-0.601 - 1.18i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (1.48 + 2.04i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.469 - 2.96i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.39 - 1.73i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.631 + 1.94i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.787 + 4.96i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-7.44 - 2.41i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.98 + 0.970i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.27 + 0.994i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (7.63 + 5.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.51 + 8.51i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.4 - 5.82i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (3.45 - 1.76i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-8.24 - 5.98i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.05 + 5.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.561 + 1.10i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (1.80 + 0.587i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 1.91i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.09 - 3.35i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.27 - 3.70i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.01 + 8.28i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.16 - 8.17i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.37786682356458899727469043656, −9.548265185441793838307065222422, −8.644110598773054101670076016149, −8.179744285944910075254942303016, −7.05400085068970667585606579887, −6.38022458006618691206346687073, −4.93707110204577278586696135741, −3.87268361188331893679175948832, −2.82275646654839566895027297834, −0.792483149175147172140218302907,
1.06892408187452921768662757010, 2.55710423617341943820173661961, 3.41037532600326280640889032702, 4.89111021887952971088487526833, 6.41252226074234369673339737291, 7.30535252133835366949377795954, 8.034444349170995583657911966481, 8.282076665293239665838769385710, 9.734613657271061928940344580027, 10.21568562527614872670427138507
