| L(s) = 1 | + (−0.186 − 0.982i)2-s + (−1.94 + 0.0455i)3-s + (−0.930 + 0.366i)4-s + (0.163 − 0.619i)5-s + (0.406 + 1.89i)6-s + (0.479 + 1.06i)7-s + (0.533 + 0.845i)8-s + (0.773 − 0.0363i)9-s + (−0.638 − 0.0449i)10-s + (−1.70 − 1.91i)11-s + (1.79 − 0.754i)12-s + (−0.242 + 3.44i)13-s + (0.959 − 0.669i)14-s + (−0.288 + 1.20i)15-s + (0.731 − 0.681i)16-s + (1.22 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.131 − 0.694i)2-s + (−1.12 + 0.0262i)3-s + (−0.465 + 0.183i)4-s + (0.0730 − 0.276i)5-s + (0.166 + 0.775i)6-s + (0.181 + 0.403i)7-s + (0.188 + 0.299i)8-s + (0.257 − 0.0121i)9-s + (−0.201 − 0.0142i)10-s + (−0.512 − 0.576i)11-s + (0.516 − 0.217i)12-s + (−0.0673 + 0.955i)13-s + (0.256 − 0.178i)14-s + (−0.0746 + 0.312i)15-s + (0.182 − 0.170i)16-s + (0.296 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.811589 - 0.119934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.811589 - 0.119934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.186 + 0.982i)T \) |
| 269 | \( 1 + (-15.6 - 4.98i)T \) |
| good | 3 | \( 1 + (1.94 - 0.0455i)T + (2.99 - 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.163 + 0.619i)T + (-4.34 - 2.46i)T^{2} \) |
| 7 | \( 1 + (-0.479 - 1.06i)T + (-4.65 + 5.23i)T^{2} \) |
| 11 | \( 1 + (1.70 + 1.91i)T + (-1.28 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.242 - 3.44i)T + (-12.8 - 1.82i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 1.08i)T + (1.98 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-6.19 - 4.11i)T + (7.37 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.138 + 5.89i)T + (-22.9 + 1.07i)T^{2} \) |
| 29 | \( 1 + (-1.20 - 2.12i)T + (-14.8 + 24.8i)T^{2} \) |
| 31 | \( 1 + (2.89 + 0.341i)T + (30.1 + 7.20i)T^{2} \) |
| 37 | \( 1 + (-1.51 + 2.82i)T + (-20.4 - 30.8i)T^{2} \) |
| 41 | \( 1 + (-11.2 - 2.12i)T + (38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-7.41 + 4.20i)T + (22.0 - 36.8i)T^{2} \) |
| 47 | \( 1 + (9.33 + 2.69i)T + (39.7 + 25.0i)T^{2} \) |
| 53 | \( 1 + (-5.62 - 7.66i)T + (-15.9 + 50.5i)T^{2} \) |
| 59 | \( 1 + (1.54 + 3.93i)T + (-43.1 + 40.2i)T^{2} \) |
| 61 | \( 1 + (-6.53 - 8.08i)T + (-12.7 + 59.6i)T^{2} \) |
| 67 | \( 1 + (0.0617 + 0.0243i)T + (49.0 + 45.6i)T^{2} \) |
| 71 | \( 1 + (7.43 - 10.6i)T + (-24.4 - 66.6i)T^{2} \) |
| 73 | \( 1 + (4.90 - 13.3i)T + (-55.6 - 47.2i)T^{2} \) |
| 79 | \( 1 + (-2.97 - 2.28i)T + (20.1 + 76.3i)T^{2} \) |
| 83 | \( 1 + (-0.981 - 10.4i)T + (-81.5 + 15.4i)T^{2} \) |
| 89 | \( 1 + (0.668 + 4.03i)T + (-84.2 + 28.6i)T^{2} \) |
| 97 | \( 1 + (11.0 - 13.6i)T + (-20.3 - 94.8i)T^{2} \) |
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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
−10.94470526808441439236302295445, −10.14967981370806264630448985955, −9.123920947763277987584316737342, −8.337632388386314407588475052079, −7.11808393723636212379943792443, −5.82114439277700761041136410330, −5.29612287501830719930034622827, −4.13429556675201460568855886441, −2.68159063899332972316948647672, −1.04528109672278420231427349826,
0.77331662405294231027607731925, 3.02826988913102483369346772415, 4.70536547554852999946689519262, 5.37147062608637782569266497739, 6.19900095753597436406807932682, 7.32566097307051639002132954641, 7.75513477546352024365714986509, 9.191232079546174985222403106811, 10.05706286967337456800362296853, 10.85014649044262985674598691880
