| L(s) = 1 | + (1.02 + 0.970i)2-s + (1.03 − 2.81i)3-s + (0.116 + 1.99i)4-s + (0.263 − 2.25i)5-s + (3.79 − 1.89i)6-s + (7.26 − 3.64i)7-s + (−1.81 + 2.16i)8-s + (−6.87 − 5.81i)9-s + (2.46 − 2.06i)10-s + (−4.66 − 2.01i)11-s + (5.74 + 1.73i)12-s + (5.88 − 1.39i)13-s + (11.0 + 3.29i)14-s + (−6.08 − 3.07i)15-s + (−3.97 + 0.464i)16-s + (20.0 − 3.53i)17-s + ⋯ |
| L(s) = 1 | + (0.514 + 0.485i)2-s + (0.343 − 0.938i)3-s + (0.0290 + 0.499i)4-s + (0.0527 − 0.451i)5-s + (0.632 − 0.316i)6-s + (1.03 − 0.521i)7-s + (−0.227 + 0.270i)8-s + (−0.763 − 0.645i)9-s + (0.246 − 0.206i)10-s + (−0.424 − 0.182i)11-s + (0.478 + 0.144i)12-s + (0.452 − 0.107i)13-s + (0.786 + 0.235i)14-s + (−0.405 − 0.204i)15-s + (−0.248 + 0.0290i)16-s + (1.18 − 0.208i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.14198 - 0.605761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14198 - 0.605761i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.02 - 0.970i)T \) |
| 3 | \( 1 + (-1.03 + 2.81i)T \) |
| good | 5 | \( 1 + (-0.263 + 2.25i)T + (-24.3 - 5.76i)T^{2} \) |
| 7 | \( 1 + (-7.26 + 3.64i)T + (29.2 - 39.3i)T^{2} \) |
| 11 | \( 1 + (4.66 + 2.01i)T + (83.0 + 88.0i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 1.39i)T + (151. - 75.8i)T^{2} \) |
| 17 | \( 1 + (-20.0 + 3.53i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (1.96 - 11.1i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (1.38 - 2.75i)T + (-315. - 424. i)T^{2} \) |
| 29 | \( 1 + (41.8 - 12.5i)T + (702. - 462. i)T^{2} \) |
| 31 | \( 1 + (20.6 - 13.5i)T + (380. - 882. i)T^{2} \) |
| 37 | \( 1 + (-39.7 - 14.4i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (37.9 - 35.7i)T + (97.7 - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-12.3 - 16.5i)T + (-530. + 1.77e3i)T^{2} \) |
| 47 | \( 1 + (-8.54 + 12.9i)T + (-874. - 2.02e3i)T^{2} \) |
| 53 | \( 1 + (30.3 - 17.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (67.0 - 28.9i)T + (2.38e3 - 2.53e3i)T^{2} \) |
| 61 | \( 1 + (4.05 - 69.6i)T + (-3.69e3 - 431. i)T^{2} \) |
| 67 | \( 1 + (-31.6 + 105. i)T + (-3.75e3 - 2.46e3i)T^{2} \) |
| 71 | \( 1 + (64.1 + 76.4i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-2.94 - 2.47i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-47.0 + 49.9i)T + (-362. - 6.23e3i)T^{2} \) |
| 83 | \( 1 + (-76.9 - 72.6i)T + (400. + 6.87e3i)T^{2} \) |
| 89 | \( 1 + (-64.7 + 77.1i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-55.9 + 6.54i)T + (9.15e3 - 2.16e3i)T^{2} \) |
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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
−12.78042824053977105063172430362, −11.83875490646194202281021738188, −10.80099612622978054498093804688, −9.106101091840206550744799256455, −7.980608033771825011769574317142, −7.52007009356888128860052345498, −6.06139139677766509760598914019, −4.98326397789331259266944890425, −3.36879999260021972601530375655, −1.42251953563405960127591571304,
2.23287981492432179577591007305, 3.59119614481440197852740056972, 4.88016343664003904268021401024, 5.79635267774751781985323272839, 7.65950868195186366801972503799, 8.798501696874216786411846639752, 9.873749243060295918291637043900, 10.90214484903997376692462436638, 11.42466406254170593173218892788, 12.70437989814185283600460219000
