| L(s) = 1 | + (0.978 + 0.207i)2-s + (−1.04 − 1.81i)3-s + (0.913 + 0.406i)4-s + (−0.0816 − 0.777i)5-s + (−0.646 − 1.99i)6-s + (0.403 − 2.61i)7-s + (0.809 + 0.587i)8-s + (−0.689 + 1.19i)9-s + (0.0816 − 0.777i)10-s + (−0.182 + 1.73i)11-s + (−0.218 − 2.08i)12-s + (−1.45 − 4.47i)13-s + (0.938 − 2.47i)14-s + (−1.32 + 0.961i)15-s + (0.669 + 0.743i)16-s + (−0.620 + 5.90i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.604 − 1.04i)3-s + (0.456 + 0.203i)4-s + (−0.0365 − 0.347i)5-s + (−0.264 − 0.812i)6-s + (0.152 − 0.988i)7-s + (0.286 + 0.207i)8-s + (−0.229 + 0.398i)9-s + (0.0258 − 0.245i)10-s + (−0.0550 + 0.523i)11-s + (−0.0631 − 0.600i)12-s + (−0.402 − 1.24i)13-s + (0.250 − 0.661i)14-s + (−0.341 + 0.248i)15-s + (0.167 + 0.185i)16-s + (−0.150 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864201 - 1.35653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864201 - 1.35653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (-0.403 + 2.61i)T \) |
| 41 | \( 1 + (-4.64 - 4.40i)T \) |
| good | 3 | \( 1 + (1.04 + 1.81i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.0816 + 0.777i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (0.182 - 1.73i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.45 + 4.47i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.620 - 5.90i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.57 + 3.96i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.22 + 0.472i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-2.40 + 1.74i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.865 + 8.23i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.666 + 6.33i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (-3.84 - 11.8i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (3.63 + 0.772i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (1.59 + 0.708i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-9.03 + 10.0i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.61 - 4.01i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (4.67 + 2.08i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-4.46 - 3.24i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.31 - 12.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.76 + 3.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + (-4.26 - 4.73i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (3.28 - 2.38i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.75174108179926211316708484866, −9.784200853123367384145843100658, −8.182882256416513854699962691789, −7.65047666834701511347187477856, −6.66637995987961996228355805264, −6.03684465339166606715188466789, −4.84440942909832082333868474667, −3.94983687199327508054557784234, −2.28287180920007511388805842155, −0.78636669569266611985690315919,
2.21079329413069078506122735841, 3.45224303571718113197299298968, 4.63016074314969836458466238718, 5.21558742342929933030755844671, 6.20563607683125035451165772372, 7.12810670882851952832918853393, 8.605066687601299278193991412180, 9.416074897478384769426230424569, 10.39159351073694232139508680942, 11.01984399423704955375513581263
