| L(s) = 1 | + (0.984 − 0.173i)2-s + (1.70 + 0.302i)3-s + (0.939 − 0.342i)4-s + (1.23 − 1.86i)5-s + (1.73 + 0.00201i)6-s + (−4.20 − 2.43i)7-s + (0.866 − 0.5i)8-s + (2.81 + 1.03i)9-s + (0.897 − 2.04i)10-s + (−0.466 + 0.269i)11-s + (1.70 − 0.298i)12-s + (3.83 + 3.21i)13-s + (−4.56 − 1.66i)14-s + (2.67 − 2.79i)15-s + (0.766 − 0.642i)16-s + (−0.787 − 4.46i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (0.984 + 0.174i)3-s + (0.469 − 0.171i)4-s + (0.554 − 0.832i)5-s + (0.707 + 0.000823i)6-s + (−1.59 − 0.918i)7-s + (0.306 − 0.176i)8-s + (0.938 + 0.344i)9-s + (0.283 − 0.647i)10-s + (−0.140 + 0.0812i)11-s + (0.492 − 0.0862i)12-s + (1.06 + 0.892i)13-s + (−1.22 − 0.444i)14-s + (0.691 − 0.722i)15-s + (0.191 − 0.160i)16-s + (−0.191 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.74747 - 1.02583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74747 - 1.02583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.70 - 0.302i)T \) |
| 5 | \( 1 + (-1.23 + 1.86i)T \) |
| 19 | \( 1 + (-2.88 - 3.26i)T \) |
| good | 7 | \( 1 + (4.20 + 2.43i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.466 - 0.269i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.83 - 3.21i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.787 + 4.46i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (3.77 - 1.37i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.161 + 0.913i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.96 - 2.86i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 + (-0.823 + 0.690i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.14 - 11.3i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.74 - 9.90i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.29 + 9.05i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0388 + 0.220i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.38 - 0.866i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.877 - 4.97i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.58 - 1.30i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.384 + 0.458i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.29 + 3.92i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.89 + 6.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.04 + 2.55i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.78 - 10.1i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.32935747666792607082748390339, −9.748017574276577966385490273286, −9.133113468291736669754916954145, −7.981530486842708127535128793714, −6.87603649339166149224419518982, −6.13558293995249254361423443086, −4.74621495515129768974192854443, −3.83747044082033110061883928087, −2.99832380285444350136041324669, −1.45347546335677629898944123409,
2.17902442367729369027339296374, 3.13802698258360618759701646055, 3.66928998408940757391070327538, 5.56811451905324600076664722821, 6.32143501167433745729460476435, 6.95727994087333418955137704788, 8.227896286302471461183293968811, 9.038931391827232905482093555558, 10.03663031529595051510703472598, 10.60113698740362223886344605884
