| L(s) = 1 | + (1.09 − 0.890i)2-s + (0.224 + 1.71i)3-s + (0.412 − 1.95i)4-s + (2.72 + 3.10i)5-s + (1.77 + 1.68i)6-s + (2.35 + 0.310i)7-s + (−1.29 − 2.51i)8-s + (−2.89 + 0.771i)9-s + (5.75 + 0.983i)10-s + (0.107 + 0.217i)11-s + (3.45 + 0.268i)12-s + (−0.686 − 2.02i)13-s + (2.86 − 1.76i)14-s + (−4.71 + 5.37i)15-s + (−3.65 − 1.61i)16-s + (−4.79 − 4.79i)17-s + ⋯ |
| L(s) = 1 | + (0.776 − 0.629i)2-s + (0.129 + 0.991i)3-s + (0.206 − 0.978i)4-s + (1.21 + 1.38i)5-s + (0.725 + 0.688i)6-s + (0.891 + 0.117i)7-s + (−0.456 − 0.889i)8-s + (−0.966 + 0.257i)9-s + (1.81 + 0.311i)10-s + (0.0322 + 0.0654i)11-s + (0.996 + 0.0775i)12-s + (−0.190 − 0.560i)13-s + (0.766 − 0.470i)14-s + (−1.21 + 1.38i)15-s + (−0.914 − 0.403i)16-s + (−1.16 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.74168 + 0.572237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74168 + 0.572237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.890i)T \) |
| 3 | \( 1 + (-0.224 - 1.71i)T \) |
| good | 5 | \( 1 + (-2.72 - 3.10i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-2.35 - 0.310i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-0.107 - 0.217i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.686 + 2.02i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.79 + 4.79i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.45 - 7.33i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.332 + 2.52i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-4.29 + 0.281i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-7.75 + 4.48i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.502 + 2.52i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.940 + 7.14i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-6.11 + 3.01i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (9.30 - 2.49i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.08 + 1.63i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.34 - 1.53i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.125 + 1.92i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-5.67 - 2.79i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.51 - 8.48i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.86 + 6.91i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.38 - 12.6i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.404 - 0.355i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (8.74 - 3.62i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (5.66 + 3.27i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.83707766971824113295369481307, −10.04310235409577080846453307144, −9.617182948711716203744620604304, −8.302060377197106121435760755253, −6.81570471566623643192379826374, −5.91823071772998213608223256386, −5.14774920631383991073382838771, −4.08778332659391821424892063759, −2.81953375781047998078740929495, −2.16268837078490598444938077076,
1.49674133412435411153613080783, 2.51247876169906961948165277439, 4.54637957294888368123548036968, 5.02495376807834379028524048799, 6.23636085901595513854940429050, 6.72139449382849482572295512823, 8.171435163298455690771321538086, 8.536902503796069400746698448295, 9.375379753610475581320461297796, 11.00477557496813615435961631535
