| L(s) = 1 | + (0.260 + 1.64i)2-s + (−2.22 − 1.13i)3-s + (−0.738 + 0.239i)4-s + (−1.39 − 1.74i)5-s + (1.28 − 3.95i)6-s + (0.297 − 0.297i)7-s + (0.925 + 1.81i)8-s + (1.89 + 2.60i)9-s + (2.51 − 2.75i)10-s + (−2.82 − 2.04i)11-s + (1.91 + 0.303i)12-s + (−0.837 + 5.29i)13-s + (0.566 + 0.411i)14-s + (1.12 + 5.46i)15-s + (−4.00 + 2.90i)16-s + (2.60 + 5.10i)17-s + ⋯ |
| L(s) = 1 | + (0.184 + 1.16i)2-s + (−1.28 − 0.653i)3-s + (−0.369 + 0.119i)4-s + (−0.623 − 0.781i)5-s + (0.524 − 1.61i)6-s + (0.112 − 0.112i)7-s + (0.327 + 0.642i)8-s + (0.631 + 0.868i)9-s + (0.794 − 0.869i)10-s + (−0.850 − 0.617i)11-s + (0.552 + 0.0874i)12-s + (−0.232 + 1.46i)13-s + (0.151 + 0.110i)14-s + (0.289 + 1.41i)15-s + (−1.00 + 0.727i)16-s + (0.631 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.146807 + 0.533906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146807 + 0.533906i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.39 + 1.74i)T \) |
| 19 | \( 1 + (-1.86 - 3.94i)T \) |
| good | 2 | \( 1 + (-0.260 - 1.64i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (2.22 + 1.13i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.297 + 0.297i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.82 + 2.04i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.837 - 5.29i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.60 - 5.10i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (0.127 + 0.803i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (1.14 + 3.52i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.78 + 2.52i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.45 + 0.388i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-4.06 - 5.59i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.56 - 6.56i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.21 + 1.63i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (6.63 + 3.38i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (2.02 - 1.47i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.23 + 5.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.28 + 3.71i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-14.0 + 4.55i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.34 - 14.7i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.724 - 2.23i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.29 - 3.20i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.684 - 0.497i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.31 - 6.50i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.31034636845473829231394833444, −10.96227578952109301636935601555, −9.409106572460097221712942096661, −8.060155036543689525995201216137, −7.72562809413291021302418501781, −6.60327618925001636838708568882, −5.84590761210690179935241900330, −5.18185179422631224976482272838, −4.09089343874379433759909791722, −1.57418802355873658933610964724,
0.37547073486752598382115184783, 2.62892715695415404959034751504, 3.53950379385974545994375604031, 4.85799404147243814473261391214, 5.48399724978392352993074015112, 7.04204339351844063672682337143, 7.62509321494557887822222811402, 9.439474392492930349920783734286, 10.31241137169209064024972686333, 10.77869801542171669807395792684
