| L(s) = 1 | + (0.204 + 1.28i)2-s + (−0.191 − 1.72i)3-s + (0.281 − 0.0915i)4-s + (−1.07 − 0.548i)5-s + (2.18 − 0.598i)6-s + (3.44 − 0.827i)7-s + (1.36 + 2.67i)8-s + (−2.92 + 0.660i)9-s + (0.487 − 1.50i)10-s + (0.784 − 0.918i)11-s + (−0.211 − 0.467i)12-s + (−0.178 − 0.291i)13-s + (1.77 + 4.27i)14-s + (−0.737 + 1.95i)15-s + (−2.68 + 1.95i)16-s + (−4.10 + 0.322i)17-s + ⋯ |
| L(s) = 1 | + (0.144 + 0.911i)2-s + (−0.110 − 0.993i)3-s + (0.140 − 0.0457i)4-s + (−0.481 − 0.245i)5-s + (0.889 − 0.244i)6-s + (1.30 − 0.312i)7-s + (0.481 + 0.944i)8-s + (−0.975 + 0.220i)9-s + (0.154 − 0.474i)10-s + (0.236 − 0.276i)11-s + (−0.0611 − 0.134i)12-s + (−0.0495 − 0.0808i)13-s + (0.473 + 1.14i)14-s + (−0.190 + 0.505i)15-s + (−0.671 + 0.487i)16-s + (−0.995 + 0.0783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21668 + 0.0872452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21668 + 0.0872452i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.191 + 1.72i)T \) |
| 41 | \( 1 + (-2.95 + 5.68i)T \) |
| good | 2 | \( 1 + (-0.204 - 1.28i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (1.07 + 0.548i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (-3.44 + 0.827i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (-0.784 + 0.918i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.178 + 0.291i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (4.10 - 0.322i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (3.81 - 6.21i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (1.27 + 0.923i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.66 - 0.445i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (3.68 + 1.19i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.78 - 5.49i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (7.41 - 1.17i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.345 + 1.43i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (1.03 - 13.1i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-3.03 + 4.17i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 11.0i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-8.31 - 9.73i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (3.32 + 2.84i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-7.90 + 7.90i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.514 + 1.24i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + (2.38 + 9.91i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.17 + 2.71i)T + (15.1 - 95.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80005061669126730872215818816, −12.40504653105301985376852026389, −11.50037131218483224884106434680, −10.70368914956637518283203147912, −8.334193051024845688530090685540, −8.096986910525149018872097324017, −6.88557087234633490021257913454, −5.87610523307562290910039683829, −4.52018428492526984709565194803, −1.88200660922536660717301022159,
2.37067556751655304189575489484, 3.97599898282920126242184546797, 4.91650388623578419034807497056, 6.80287708106716390458931212167, 8.282915670347991737134775688481, 9.429032785363665608863928101879, 10.74101791587960643155270600980, 11.27831431536482980950519545593, 11.86690952922977929881273804453, 13.24137563394942619796104588083
