| L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s − 3.02·11-s + (2.44 − 2.44i)12-s + (−2.99 − 2.99i)13-s + 3.74i·14-s − 4·16-s + (13.0 − 13.0i)17-s + (2.99 + 2.99i)18-s − 29.5i·19-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s − 0.274·11-s + (0.204 − 0.204i)12-s + (−0.230 − 0.230i)13-s + 0.267i·14-s − 0.250·16-s + (0.768 − 0.768i)17-s + (0.166 + 0.166i)18-s − 1.55i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.527823209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527823209\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
| good | 11 | \( 1 + 3.02T + 121T^{2} \) |
| 13 | \( 1 + (2.99 + 2.99i)T + 169iT^{2} \) |
| 17 | \( 1 + (-13.0 + 13.0i)T - 289iT^{2} \) |
| 19 | \( 1 + 29.5iT - 361T^{2} \) |
| 23 | \( 1 + (-2.34 - 2.34i)T + 529iT^{2} \) |
| 29 | \( 1 + 24.4iT - 841T^{2} \) |
| 31 | \( 1 - 46.7T + 961T^{2} \) |
| 37 | \( 1 + (-30.6 + 30.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 11.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (0.402 + 0.402i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-46.4 + 46.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-11.9 - 11.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 32.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (17.3 - 17.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 43.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-39.4 - 39.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 0.100iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (15.9 + 15.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 91.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-14.1 + 14.1i)T - 9.40e3iT^{2} \) |
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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
−9.613792657030627081405934968838, −8.965658782885510656854827948861, −7.908065324251235663535786492685, −7.00209415806933682238573030494, −5.88060713725041557088075108700, −5.01858553335785754874960230877, −4.21375900649291110420612283657, −3.00049502515479522384554336814, −2.44694951846376066981225443970, −0.66204799252047195649048382416,
1.33277433535818801258278292828, 2.76408212666854383741654534618, 3.68735295651884381779168578991, 4.64679385085526508410327395258, 5.85846373818538058473605598537, 6.42100218552709864627322944358, 7.52986072584436940808638501180, 7.981500135098552937073994725874, 8.875906398044594675124585382247, 9.917574771202498151000632949691
