| 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
−11.64553176850661424260614333554, −10.62965585574861497340661598837, −9.727919335804390055058005735540, −8.954555105505838521566129004680, −7.891197533321847021772082519756, −6.79900087482997058150649406434, −5.82342423897237659863498796213, −4.48382865385981964113808468161, −3.04074976899952369143935965326, −1.62971731030974435480704259421,
1.95999729966454278979058544130, 3.12998570566846868476128448405, 4.74577587145520240758053276057, 5.81375012472657840532456203528, 6.84441482001603456720328655706, 8.255477612973095427462171235377, 8.876522770944674427288517424722, 10.08613719036699661613602992733, 10.44709684572760166209795070710, 11.91779991795138463487640252486
