| L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.0523 + 0.998i)3-s + (−0.587 − 0.809i)4-s + (1.31 + 1.80i)5-s + (0.866 + 0.5i)6-s + (1.60 − 1.98i)7-s + (−0.987 + 0.156i)8-s + (−0.994 − 0.104i)9-s + (2.20 − 0.351i)10-s + (1.20 − 2.70i)11-s + (0.838 − 0.544i)12-s + (2.22 + 1.44i)13-s + (−1.03 − 2.33i)14-s + (−1.87 + 1.21i)15-s + (−0.309 + 0.951i)16-s + (1.26 − 3.30i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (−0.0302 + 0.576i)3-s + (−0.293 − 0.404i)4-s + (0.588 + 0.808i)5-s + (0.353 + 0.204i)6-s + (0.607 − 0.750i)7-s + (−0.349 + 0.0553i)8-s + (−0.331 − 0.0348i)9-s + (0.698 − 0.111i)10-s + (0.363 − 0.815i)11-s + (0.242 − 0.157i)12-s + (0.617 + 0.401i)13-s + (−0.277 − 0.624i)14-s + (−0.483 + 0.314i)15-s + (−0.0772 + 0.237i)16-s + (0.307 − 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.14045 - 0.468249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14045 - 0.468249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 + 0.891i)T \) |
| 3 | \( 1 + (0.0523 - 0.998i)T \) |
| 5 | \( 1 + (-1.31 - 1.80i)T \) |
| 31 | \( 1 + (-5.13 + 2.14i)T \) |
| good | 7 | \( 1 + (-1.60 + 1.98i)T + (-1.45 - 6.84i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 2.70i)T + (-7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.22 - 1.44i)T + (5.28 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 3.30i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.221 - 1.04i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.103 + 0.652i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.30 - 4.02i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.907 - 3.38i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.06 + 3.39i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (0.122 + 0.188i)T + (-17.4 + 39.2i)T^{2} \) |
| 47 | \( 1 + (-9.60 + 4.89i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (2.67 - 2.16i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (5.39 - 4.85i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 1.69iT - 61T^{2} \) |
| 67 | \( 1 + (1.94 - 7.25i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.805 + 7.66i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.312 - 0.814i)T + (-54.2 + 48.8i)T^{2} \) |
| 79 | \( 1 + (-6.92 + 3.08i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (15.8 - 0.829i)T + (82.5 - 8.67i)T^{2} \) |
| 89 | \( 1 + (-5.20 + 3.78i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.535 - 0.0847i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.26065800836348062523933675771, −9.393972787776215453062816583873, −8.586704773910151860831193856981, −7.41767261835040487177409083312, −6.38592949190531967303666108650, −5.58109619054899291390803329361, −4.49514640760910934692426242547, −3.62160709595527776069650952620, −2.68487587847039193807518314266, −1.21237192606332944564561074056,
1.32243220246425125515148361852, 2.48587899230689773968467901076, 4.11715914862490065457402402570, 5.05508714109257223308331562872, 5.85700061600534914965729243318, 6.48595621147243596535706468696, 7.71608448450373530799603546058, 8.335162380864650787492376166887, 9.036449185090133019372048885541, 9.901596150002794071169563385461
