L(s) = 1 | + (0.951 + 0.309i)3-s + (2.23 + 0.0974i)5-s + 1.31i·7-s + (0.809 + 0.587i)9-s + (−1.25 + 0.913i)11-s + (1.42 − 1.96i)13-s + (2.09 + 0.782i)15-s + (−1.25 + 0.406i)17-s + (0.315 + 0.971i)19-s + (−0.407 + 1.25i)21-s + (−2.94 − 4.05i)23-s + (4.98 + 0.435i)25-s + (0.587 + 0.809i)27-s + (1.82 − 5.61i)29-s + (2.73 + 8.41i)31-s + ⋯ |
L(s) = 1 | + (0.549 + 0.178i)3-s + (0.999 + 0.0435i)5-s + 0.498i·7-s + (0.269 + 0.195i)9-s + (−0.379 + 0.275i)11-s + (0.395 − 0.544i)13-s + (0.540 + 0.202i)15-s + (−0.303 + 0.0985i)17-s + (0.0723 + 0.222i)19-s + (−0.0889 + 0.273i)21-s + (−0.613 − 0.844i)23-s + (0.996 + 0.0870i)25-s + (0.113 + 0.155i)27-s + (0.338 − 1.04i)29-s + (0.491 + 1.51i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72145 + 0.306731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72145 + 0.306731i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.23 - 0.0974i)T \) |
good | 7 | \( 1 - 1.31iT - 7T^{2} \) |
| 11 | \( 1 + (1.25 - 0.913i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 1.96i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.25 - 0.406i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.315 - 0.971i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.94 + 4.05i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.82 + 5.61i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.73 - 8.41i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.95 - 4.06i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.43 + 4.67i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.84iT - 43T^{2} \) |
| 47 | \( 1 + (7.37 + 2.39i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.75 + 1.22i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.35 + 4.61i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 2.05i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (7.92 - 2.57i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.00 - 12.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.47 + 10.2i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.386 + 1.18i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.80 - 2.53i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 5.62i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (15.4 + 5.02i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.91779991795138463487640252486, −10.44709684572760166209795070710, −10.08613719036699661613602992733, −8.876522770944674427288517424722, −8.255477612973095427462171235377, −6.84441482001603456720328655706, −5.81375012472657840532456203528, −4.74577587145520240758053276057, −3.12998570566846868476128448405, −1.95999729966454278979058544130,
1.62971731030974435480704259421, 3.04074976899952369143935965326, 4.48382865385981964113808468161, 5.82342423897237659863498796213, 6.79900087482997058150649406434, 7.891197533321847021772082519756, 8.954555105505838521566129004680, 9.727919335804390055058005735540, 10.62965585574861497340661598837, 11.64553176850661424260614333554