| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.720 + 2.21i)3-s + (0.309 − 0.951i)4-s + (−2.02 + 0.951i)5-s + (−0.720 − 2.21i)6-s + 3.77·7-s + (0.309 + 0.951i)8-s + (−1.97 − 1.43i)9-s + (1.07 − 1.95i)10-s + (3.05 − 2.21i)11-s + (1.88 + 1.37i)12-s + (−2.56 − 1.86i)13-s + (−3.05 + 2.21i)14-s + (−0.651 − 5.17i)15-s + (−0.809 − 0.587i)16-s + (−0.430 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.416 + 1.28i)3-s + (0.154 − 0.475i)4-s + (−0.905 + 0.425i)5-s + (−0.294 − 0.905i)6-s + 1.42·7-s + (0.109 + 0.336i)8-s + (−0.658 − 0.478i)9-s + (0.340 − 0.619i)10-s + (0.920 − 0.668i)11-s + (0.544 + 0.395i)12-s + (−0.712 − 0.517i)13-s + (−0.816 + 0.592i)14-s + (−0.168 − 1.33i)15-s + (−0.202 − 0.146i)16-s + (−0.104 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0617 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0617 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.398635 + 0.424077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398635 + 0.424077i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (2.02 - 0.951i)T \) |
| good | 3 | \( 1 + (0.720 - 2.21i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + (-3.05 + 2.21i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.56 + 1.86i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.430 + 1.32i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.20 - 3.72i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.720 - 0.523i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0152 + 0.0468i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.70 + 4.14i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 0.875i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 + (-1.16 + 3.58i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.58 + 11.0i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.558 - 0.405i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.38 - 6.08i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.73 - 14.5i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.06 - 6.36i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.18 - 3.03i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.558 - 1.71i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.08 + 9.47i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (11.7 - 8.52i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.0278 - 0.0857i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−15.94066403816652874539645046360, −14.91552700982765884373941443955, −14.36892150312208587938842387115, −11.77554873709456859688976461371, −11.11781135060770340809757695786, −10.06864390417272949242243837319, −8.624770690441545177914175786236, −7.45305952338478573724332821913, −5.44942981255542042506670333758, −4.07189313674011906234463287729,
1.56569813741413901498536944118, 4.61552036778177781032421415337, 6.96858237703046381466001458109, 7.79644492143105854784281877797, 9.017311440686870415391501683068, 11.04553720799934630366569332777, 11.95146064190811267195375643880, 12.42398089260764695306428084833, 14.02992016192811668472495079877, 15.31148280886429014681585907214
