| L(s) = 1 | + (−0.976 + 0.214i)2-s + (0.947 + 1.44i)3-s + (0.907 − 0.419i)4-s + (−1.88 − 0.524i)5-s + (−1.23 − 1.21i)6-s + (0.743 − 0.704i)7-s + (−0.796 + 0.605i)8-s + (−1.20 + 2.74i)9-s + (1.95 + 0.106i)10-s + (0.616 + 0.726i)11-s + (1.46 + 0.917i)12-s + (2.28 + 6.77i)13-s + (−0.574 + 0.847i)14-s + (−1.03 − 3.23i)15-s + (0.647 − 0.762i)16-s + (−2.91 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.152i)2-s + (0.547 + 0.837i)3-s + (0.453 − 0.209i)4-s + (−0.845 − 0.234i)5-s + (−0.505 − 0.494i)6-s + (0.280 − 0.266i)7-s + (−0.281 + 0.213i)8-s + (−0.401 + 0.916i)9-s + (0.619 + 0.0335i)10-s + (0.185 + 0.218i)11-s + (0.424 + 0.264i)12-s + (0.633 + 1.87i)13-s + (−0.153 + 0.226i)14-s + (−0.266 − 0.835i)15-s + (0.161 − 0.190i)16-s + (−0.706 + 0.745i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.546879 + 0.757278i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546879 + 0.757278i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.976 - 0.214i)T \) |
| 3 | \( 1 + (-0.947 - 1.44i)T \) |
| 59 | \( 1 + (-5.74 + 5.10i)T \) |
| good | 5 | \( 1 + (1.88 + 0.524i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.743 + 0.704i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.616 - 0.726i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.28 - 6.77i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (2.91 - 3.07i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.60 - 4.03i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 0.152i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-0.263 + 1.19i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (4.72 + 1.88i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-5.87 + 7.72i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-0.0278 + 0.256i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (1.05 + 0.892i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-3.52 - 12.6i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-8.90 + 0.482i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (1.02 + 4.63i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (5.80 + 7.63i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-8.61 + 2.39i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (0.629 + 0.427i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (0.201 + 1.23i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-2.69 - 5.07i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-1.01 - 0.223i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (-8.25 + 5.59i)T + (35.9 - 90.1i)T^{2} \) |
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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
−11.29888840683048363246152576568, −10.92709323216121127013948723075, −9.638224958847585303157239000297, −9.008571596259359067724716359624, −8.168795999765994166154858953689, −7.35037872133843540413963676084, −6.04972253185905604863836245781, −4.41110257925650726601046580543, −3.83675066468833619954814868046, −1.93321758436714023949886528413,
0.77839103556677856735906165033, 2.62417591179137100196006119842, 3.56701137919231971034972946834, 5.46593803976285356877340384764, 6.82372083891615786101226137552, 7.56337522365640186987041765586, 8.384546015310605544412993297293, 8.997105763324860439076430773018, 10.30572691953707416291840131356, 11.38722833215925938808453726843
