| L(s) = 1 | + (2.44 + 0.213i)2-s + (−1.61 + 0.617i)3-s + (3.96 + 0.698i)4-s + (1.02 + 1.98i)5-s + (−4.08 + 1.16i)6-s + (−3.87 − 2.71i)7-s + (4.80 + 1.28i)8-s + (2.23 − 1.99i)9-s + (2.07 + 5.08i)10-s + (0.927 − 2.54i)11-s + (−6.84 + 1.31i)12-s + (0.0885 + 1.01i)13-s + (−8.89 − 7.46i)14-s + (−2.88 − 2.58i)15-s + (3.89 + 1.41i)16-s + (−1.15 + 0.309i)17-s + ⋯ |
| L(s) = 1 | + (1.72 + 0.151i)2-s + (−0.934 + 0.356i)3-s + (1.98 + 0.349i)4-s + (0.457 + 0.889i)5-s + (−1.66 + 0.475i)6-s + (−1.46 − 1.02i)7-s + (1.69 + 0.454i)8-s + (0.745 − 0.666i)9-s + (0.655 + 1.60i)10-s + (0.279 − 0.768i)11-s + (−1.97 + 0.380i)12-s + (0.0245 + 0.280i)13-s + (−2.37 − 1.99i)14-s + (−0.744 − 0.667i)15-s + (0.974 + 0.354i)16-s + (−0.279 + 0.0749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89656 + 0.534984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89656 + 0.534984i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.61 - 0.617i)T \) |
| 5 | \( 1 + (-1.02 - 1.98i)T \) |
| good | 2 | \( 1 + (-2.44 - 0.213i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (3.87 + 2.71i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.927 + 2.54i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0885 - 1.01i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.15 - 0.309i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.507 + 0.292i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.750 + 1.07i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.185 - 0.155i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.978 - 5.54i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.227 + 0.850i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.80 - 3.33i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.67 - 10.0i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-3.77 + 5.39i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-6.73 + 6.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (-11.1 + 4.07i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.40 + 7.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.77 - 0.680i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-1.13 - 0.654i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.567 - 2.11i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 2.59i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.839 - 9.59i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (1.54 + 2.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.04 - 1.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.37977864911286618913496672717, −12.57146114393721176667911140213, −11.39747731749386483926873270549, −10.65275985277604643648219813507, −9.669011553411992141664695784616, −6.93609612573579957677487519059, −6.56254315707161446036173063539, −5.61410982407979148601220782646, −4.07409137570156564222468577036, −3.19528975353270919855078828900,
2.31430278873036571518349491154, 4.16861420152601051772705573300, 5.48197292808267961876368536911, 5.97684842879085588736994764293, 7.06496127762850389753925666571, 9.192758267760172593209194406597, 10.35014575280754476456416707195, 11.93735642795219713746579885375, 12.23291919407980494180499444111, 13.05434556870688934817560009413
