| L(s) = 1 | + (−1.23 + 0.680i)2-s + (−0.975 + 1.43i)3-s + (1.07 − 1.68i)4-s + (0.356 − 0.129i)5-s + (0.235 − 2.43i)6-s + (−1.66 + 0.961i)7-s + (−0.181 + 2.82i)8-s + (−1.09 − 2.79i)9-s + (−0.353 + 0.403i)10-s + (0.285 + 0.164i)11-s + (1.36 + 3.18i)12-s + (−2.20 − 2.63i)13-s + (1.40 − 2.32i)14-s + (−0.162 + 0.637i)15-s + (−1.69 − 3.62i)16-s + (−4.48 − 0.790i)17-s + ⋯ |
| L(s) = 1 | + (−0.876 + 0.481i)2-s + (−0.563 + 0.826i)3-s + (0.536 − 0.843i)4-s + (0.159 − 0.0580i)5-s + (0.0961 − 0.995i)6-s + (−0.629 + 0.363i)7-s + (−0.0641 + 0.997i)8-s + (−0.365 − 0.930i)9-s + (−0.111 + 0.127i)10-s + (0.0861 + 0.0497i)11-s + (0.394 + 0.918i)12-s + (−0.612 − 0.729i)13-s + (0.376 − 0.621i)14-s + (−0.0419 + 0.164i)15-s + (−0.424 − 0.905i)16-s + (−1.08 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0891 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0891 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148379 - 0.135690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148379 - 0.135690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 0.680i)T \) |
| 3 | \( 1 + (0.975 - 1.43i)T \) |
| 19 | \( 1 + (-3.91 - 1.91i)T \) |
| good | 5 | \( 1 + (-0.356 + 0.129i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.66 - 0.961i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.285 - 0.164i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.20 + 2.63i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.48 + 0.790i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (6.59 + 2.40i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.57 + 8.91i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.02 + 1.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.18iT - 37T^{2} \) |
| 41 | \( 1 + (-6.03 + 7.19i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.50 + 0.546i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.36 + 7.73i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.94 - 1.43i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-3.81 - 0.672i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 3.18i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.260 + 1.47i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (11.1 - 4.05i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (9.19 + 7.71i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.14 - 8.50i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (12.4 - 7.16i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.50 - 5.36i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.00 - 11.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.55556756099970683102020858621, −9.780057945119261156778396325761, −9.369778130057136881337779248851, −8.254776644125220545968579400281, −7.17189489613661508076301108283, −6.00107404792917502356184618554, −5.54835028419256893857308729583, −4.13238044951345634007646004680, −2.47022574132251536661411713671, −0.17119066209395115080008954206,
1.58247694705993822372344411856, 2.83125857141840886891046036115, 4.38808349692213331411721155857, 6.02988283207666941858823416879, 6.87200737562472507343179139134, 7.52868064828776013905284206561, 8.616758309696194687966148755372, 9.624112997922657475440776426906, 10.34991103525571928701192265157, 11.41321707687882022184772465898
