| L(s) = 1 | + (−0.0240 − 0.0991i)2-s + (0.327 + 0.945i)3-s + (1.76 − 0.911i)4-s + (−2.72 + 1.09i)5-s + (0.0858 − 0.0551i)6-s + (−0.0983 + 2.64i)7-s + (−0.266 − 0.307i)8-s + (−0.786 + 0.618i)9-s + (0.174 + 0.244i)10-s + (−0.169 + 0.700i)11-s + (1.43 + 1.37i)12-s + (−0.589 + 1.29i)13-s + (0.264 − 0.0538i)14-s + (−1.92 − 2.22i)15-s + (2.28 − 3.20i)16-s + (0.239 + 5.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.0170 − 0.0701i)2-s + (0.188 + 0.545i)3-s + (0.884 − 0.455i)4-s + (−1.22 + 0.488i)5-s + (0.0350 − 0.0225i)6-s + (−0.0371 + 0.999i)7-s + (−0.0942 − 0.108i)8-s + (−0.262 + 0.206i)9-s + (0.0550 + 0.0772i)10-s + (−0.0512 + 0.211i)11-s + (0.415 + 0.396i)12-s + (−0.163 + 0.357i)13-s + (0.0707 − 0.0143i)14-s + (−0.497 − 0.573i)15-s + (0.571 − 0.801i)16-s + (0.0581 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.847159 + 0.951555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847159 + 0.951555i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.327 - 0.945i)T \) |
| 7 | \( 1 + (0.0983 - 2.64i)T \) |
| 23 | \( 1 + (4.70 - 0.921i)T \) |
| good | 2 | \( 1 + (0.0240 + 0.0991i)T + (-1.77 + 0.916i)T^{2} \) |
| 5 | \( 1 + (2.72 - 1.09i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (0.169 - 0.700i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.589 - 1.29i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.239 - 5.03i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (0.191 - 4.01i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (-1.87 + 1.20i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-9.14 + 1.76i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-0.290 + 0.228i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.912 + 6.34i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.97 - 3.42i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-3.35 - 5.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.806 + 0.0769i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (-1.19 - 1.68i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (0.362 - 1.04i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (-11.3 + 10.8i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-6.10 - 1.79i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (9.02 - 4.65i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (-6.19 + 0.591i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (1.33 + 9.31i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-14.4 - 2.77i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (0.205 - 1.42i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.24666499936341344291502734384, −10.39012482426937337994393215264, −9.663768664236141822953073167832, −8.340500312587539212155802830001, −7.76816634679400554901508162931, −6.51359142144974597846576420701, −5.74264507241550356182729630502, −4.32091579725733162127821865370, −3.27344466414795925227069775958, −2.09181530060306920801691421152,
0.74805905998656822522243461629, 2.68095550254429999383124313190, 3.73983612553149956530187013852, 4.85942971670661154519509543076, 6.50342604681439227907896153471, 7.21108484991504491009390143368, 7.921396713575921658367454537921, 8.507297050537638515295727447106, 9.955698041271523513267308284271, 11.01579884401735369714636058157
