| L(s) = 1 | + (1.16 + 0.805i)2-s + (1.05 + 1.93i)3-s + (0.703 + 1.87i)4-s + (1.36 − 1.77i)5-s + (−0.330 + 3.10i)6-s + (0.469 + 0.626i)7-s + (−0.689 + 2.74i)8-s + (−1.01 + 1.57i)9-s + (3.01 − 0.965i)10-s + (0.247 − 0.113i)11-s + (−2.88 + 3.34i)12-s + (−3.81 − 2.85i)13-s + (0.0408 + 1.10i)14-s + (4.87 + 0.762i)15-s + (−3.01 + 2.63i)16-s + (3.13 − 0.224i)17-s + ⋯ |
| L(s) = 1 | + (0.822 + 0.569i)2-s + (0.610 + 1.11i)3-s + (0.351 + 0.936i)4-s + (0.608 − 0.793i)5-s + (−0.134 + 1.26i)6-s + (0.177 + 0.236i)7-s + (−0.243 + 0.969i)8-s + (−0.337 + 0.525i)9-s + (0.952 − 0.305i)10-s + (0.0747 − 0.0341i)11-s + (−0.832 + 0.965i)12-s + (−1.05 − 0.791i)13-s + (0.0109 + 0.295i)14-s + (1.25 + 0.196i)15-s + (−0.752 + 0.658i)16-s + (0.760 − 0.0543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0761 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0761 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92261 + 2.07508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92261 + 2.07508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 - 0.805i)T \) |
| 5 | \( 1 + (-1.36 + 1.77i)T \) |
| 23 | \( 1 + (4.69 + 0.985i)T \) |
| good | 3 | \( 1 + (-1.05 - 1.93i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-0.469 - 0.626i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-0.247 + 0.113i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (3.81 + 2.85i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 0.224i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (0.751 + 0.866i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.31 - 1.14i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.79 + 6.11i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.595 - 2.73i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (5.60 - 3.60i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.89 - 5.29i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (5.11 - 5.11i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.98 + 7.99i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (0.977 + 6.79i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (0.633 - 0.186i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.98 - 10.6i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-9.86 - 4.50i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-13.9 - 0.999i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.0246 - 0.171i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-1.45 - 6.69i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-4.08 + 13.9i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-8.79 - 1.91i)T + (88.2 + 40.2i)T^{2} \) |
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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
−11.43181730154470470809664740325, −10.05581050703052391638782734669, −9.585550728981700297050319452277, −8.456170832104229571172817246301, −7.86790009213002259666278684567, −6.37358117419733947137716556615, −5.27297620917136313870072290767, −4.71888622102262298640548789777, −3.60186568536241265400846840700, −2.42139051739816227179296860693,
1.66811454153821380134691235469, 2.43072858830933599197463237390, 3.60959066061579751107320864216, 5.05360123317218871810519789080, 6.25286767101023027214441895138, 7.00848248299926227466592144057, 7.75657640316456504080650585376, 9.229615333436400500402297461603, 10.11303730010679427733844727226, 10.87573945270517277169476354824
