| L(s) = 1 | + (0.728 − 0.684i)2-s + (1.31 + 2.07i)3-s + (0.0627 − 0.998i)4-s + (1.03 − 1.98i)5-s + (2.38 + 0.612i)6-s + (1.35 − 4.18i)7-s + (−0.637 − 0.770i)8-s + (−1.30 + 2.77i)9-s + (−0.604 − 2.15i)10-s + (−4.26 + 4.00i)11-s + (2.15 − 1.18i)12-s + (−1.15 + 2.45i)13-s + (−1.87 − 3.97i)14-s + (5.48 − 0.467i)15-s + (−0.992 − 0.125i)16-s + (0.120 + 1.90i)17-s + ⋯ |
| L(s) = 1 | + (0.515 − 0.484i)2-s + (0.761 + 1.20i)3-s + (0.0313 − 0.499i)4-s + (0.462 − 0.886i)5-s + (0.973 + 0.250i)6-s + (0.513 − 1.58i)7-s + (−0.225 − 0.272i)8-s + (−0.435 + 0.924i)9-s + (−0.191 − 0.680i)10-s + (−1.28 + 1.20i)11-s + (0.623 − 0.342i)12-s + (−0.320 + 0.680i)13-s + (−0.500 − 1.06i)14-s + (1.41 − 0.120i)15-s + (−0.248 − 0.0313i)16-s + (0.0291 + 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03947 - 0.358584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03947 - 0.358584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.728 + 0.684i)T \) |
| 5 | \( 1 + (-1.03 + 1.98i)T \) |
| good | 3 | \( 1 + (-1.31 - 2.07i)T + (-1.27 + 2.71i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 4.18i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (4.26 - 4.00i)T + (0.690 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.15 - 2.45i)T + (-8.28 - 10.0i)T^{2} \) |
| 17 | \( 1 + (-0.120 - 1.90i)T + (-16.8 + 2.13i)T^{2} \) |
| 19 | \( 1 + (0.345 - 0.544i)T + (-8.08 - 17.1i)T^{2} \) |
| 23 | \( 1 + (1.17 - 6.14i)T + (-21.3 - 8.46i)T^{2} \) |
| 29 | \( 1 + (-4.60 - 1.82i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (-0.185 - 2.95i)T + (-30.7 + 3.88i)T^{2} \) |
| 37 | \( 1 + (7.11 + 0.898i)T + (35.8 + 9.20i)T^{2} \) |
| 41 | \( 1 + (1.67 + 8.77i)T + (-38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-1.07 - 0.784i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (4.03 - 4.87i)T + (-8.80 - 46.1i)T^{2} \) |
| 53 | \( 1 + (-11.0 + 2.84i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (-4.53 + 2.49i)T + (31.6 - 49.8i)T^{2} \) |
| 61 | \( 1 + (-2.63 + 13.7i)T + (-56.7 - 22.4i)T^{2} \) |
| 67 | \( 1 + (2.62 - 1.03i)T + (48.8 - 45.8i)T^{2} \) |
| 71 | \( 1 + (-5.38 + 6.51i)T + (-13.3 - 69.7i)T^{2} \) |
| 73 | \( 1 + (5.34 + 2.94i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (3.63 + 5.73i)T + (-33.6 + 71.4i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 2.06i)T + (-35.3 - 75.1i)T^{2} \) |
| 89 | \( 1 + (7.49 + 4.11i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (-1.91 - 0.758i)T + (70.7 + 66.4i)T^{2} \) |
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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
−12.10632211802736254316280948321, −10.64019050702495070380755274462, −10.19012908902768948686619681124, −9.474198030485901875686683074636, −8.289397924885807533768671874362, −7.15516012549274446799377029903, −5.14591254236991656631708599685, −4.57314031822810919921056848716, −3.67025103419641560507528439874, −1.89253339676648308761020535082,
2.52494322103010450379624186341, 2.79847067714947816454049247302, 5.28979519984352239402409738876, 6.05513706046143834919553871873, 7.12963324531264164253405638557, 8.247183021713579638823254690227, 8.558574874997754701519105568608, 10.23915882382514441051597037303, 11.47174872424806354792999315811, 12.37116806639264475848769917926
