| L(s) = 1 | + (−1.22 − 1.22i)3-s + 2.99i·9-s + (4.89 − 4.89i)17-s + 4i·19-s + (−2.44 − 2.44i)23-s + (3.67 − 3.67i)27-s + 8·31-s + (7.34 − 7.34i)47-s − 7i·49-s − 11.9·51-s + (−9.79 − 9.79i)53-s + (4.89 − 4.89i)57-s + 2·61-s + 5.99i·69-s − 16i·79-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)3-s + 0.999i·9-s + (1.18 − 1.18i)17-s + 0.917i·19-s + (−0.510 − 0.510i)23-s + (0.707 − 0.707i)27-s + 1.43·31-s + (1.07 − 1.07i)47-s − i·49-s − 1.68·51-s + (−1.34 − 1.34i)53-s + (0.648 − 0.648i)57-s + 0.256·61-s + 0.722i·69-s − 1.80i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182381584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182381584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-4.89 + 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 + 2.44i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.79 + 9.79i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 16iT - 79T^{2} \) |
| 83 | \( 1 + (2.44 + 2.44i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−9.857985006598234488345666815105, −8.543750260145500196146381658716, −7.83738402748814891889678700861, −7.08558327658356964980663191477, −6.21576872336651521318688874211, −5.45014106752503096341868284215, −4.59657720775064796326895936823, −3.24587447288801895800646260029, −2.00221240568426691421000120027, −0.67146782116457868039666048247,
1.14226870240276052707038592325, 2.90068815041112368901336445051, 3.95500394146663808701560697744, 4.76081794227173862564391240751, 5.77290692144368397795452045057, 6.31986026313351043401535364921, 7.45525067452574094576175693420, 8.347218656624494002227588958727, 9.321476573187891995132217725320, 9.963721228259581651547765615741
