L(s) = 1 | + (−0.766 + 0.642i)2-s + (−2.56 − 0.934i)3-s + (0.173 − 0.984i)4-s + (2.56 − 0.934i)6-s + (1.06 − 1.85i)7-s + (0.500 + 0.866i)8-s + (3.42 + 2.87i)9-s + (0.926 + 1.60i)11-s + (−1.36 + 2.36i)12-s + (−5.18 + 1.88i)13-s + (0.371 + 2.10i)14-s + (−0.939 − 0.342i)16-s + (2.82 − 2.37i)17-s − 4.46·18-s + (1.21 + 4.18i)19-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (−1.48 − 0.539i)3-s + (0.0868 − 0.492i)4-s + (1.04 − 0.381i)6-s + (0.403 − 0.699i)7-s + (0.176 + 0.306i)8-s + (1.14 + 0.957i)9-s + (0.279 + 0.483i)11-s + (−0.394 + 0.683i)12-s + (−1.43 + 0.522i)13-s + (0.0991 + 0.562i)14-s + (−0.234 − 0.0855i)16-s + (0.686 − 0.575i)17-s − 1.05·18-s + (0.278 + 0.960i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.628407 + 0.0290803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.628407 + 0.0290803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-1.21 - 4.18i)T \) |
good | 3 | \( 1 + (2.56 + 0.934i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 1.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.926 - 1.60i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.18 - 1.88i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 2.37i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.511 - 2.89i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.36 + 4.50i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.12 + 3.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + (-1.47 - 0.536i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.404 - 2.29i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.59 - 7.20i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.11 + 6.32i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 9.07i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0461 - 0.261i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 9.49i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.722 - 4.09i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-14.4 - 5.27i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 0.985i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.00 + 12.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.86 - 0.676i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.28 + 6.94i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.892183219206808745849764697317, −9.603838350112949285075765081080, −7.958026806884110298966150937549, −7.37006948998311891946376264436, −6.83395029475577516340990577087, −5.77341394852931520467522953944, −5.12610855361865690381200723761, −4.13336045983958901136398807148, −1.99611163874847538655009328614, −0.76625407806191501190598262526,
0.69122229959828286234616024135, 2.37632222203099904854970826717, 3.75190705607989288075196449066, 5.06430233036087865168340369865, 5.40544108305987504537698102955, 6.56691367171163062746500683493, 7.47135360787298222168388712072, 8.586118096731687947890215601519, 9.383404204820458305873203332008, 10.28155995319828376608399061090