| L(s) = 1 | + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + 2.44i·6-s + (3.97 − 5.76i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (−1.64 − 2.85i)11-s + (−2.99 − 1.73i)12-s + 7.72i·13-s + (4.24 + 8.94i)14-s + (−2.00 + 3.46i)16-s + (10.9 − 6.30i)17-s + (2.12 + 3.67i)18-s + (1.54 + 0.890i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + 0.408i·6-s + (0.567 − 0.823i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.149 − 0.259i)11-s + (−0.249 − 0.144i)12-s + 0.594i·13-s + (0.303 + 0.638i)14-s + (−0.125 + 0.216i)16-s + (0.642 − 0.371i)17-s + (0.117 + 0.204i)18-s + (0.0812 + 0.0468i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.890592045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890592045\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-3.97 + 5.76i)T \) |
| good | 11 | \( 1 + (1.64 + 2.85i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 7.72iT - 169T^{2} \) |
| 17 | \( 1 + (-10.9 + 6.30i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 0.890i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (3.37 - 5.85i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 39.2T + 841T^{2} \) |
| 31 | \( 1 + (-9.46 + 5.46i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-17.1 + 29.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 77.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (11.1 + 6.44i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (25.8 + 44.7i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-97.7 + 56.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.7 + 13.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (9.95 + 17.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 87.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (52.7 - 30.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.7 + 30.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (47.4 + 27.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 45.7iT - 9.40e3T^{2} \) |
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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
−9.613025819261217829448568501661, −8.496367106490413998425347235731, −8.047360442265622536108071411703, −7.16626910568423077136604757569, −6.54743025531875093310363982722, −5.34997440685156411958552963894, −4.43198555916513927795365000948, −3.35450334411593404125946415252, −1.87820510234246171295027343217, −0.68878256867370484072676357753,
1.26766573855632844880985546026, 2.47429266993349198028244250264, 3.25061669632166059958090220011, 4.50172159728035875866835255634, 5.28140718062339910884430505776, 6.45725626818982612562855395399, 7.77009115141785928972592927233, 8.285555591175444044857906665062, 8.953188737405175961775274456786, 9.985600655484381232580778980062
