L(s) = 1 | + (−0.232 + 0.402i)3-s + (0.5 − 0.866i)5-s + 4.83·7-s + (1.39 + 2.41i)9-s + 2.46·11-s + (2.23 + 3.87i)13-s + (0.232 + 0.402i)15-s + (−1.30 + 2.26i)17-s + (0.851 − 4.27i)19-s + (−1.12 + 1.94i)21-s + (−0.927 − 1.60i)23-s + (−0.499 − 0.866i)25-s − 2.68·27-s + (2.19 + 3.80i)29-s − 8.61·31-s + ⋯ |
L(s) = 1 | + (−0.134 + 0.232i)3-s + (0.223 − 0.387i)5-s + 1.82·7-s + (0.464 + 0.803i)9-s + 0.743·11-s + (0.620 + 1.07i)13-s + (0.0599 + 0.103i)15-s + (−0.317 + 0.549i)17-s + (0.195 − 0.980i)19-s + (−0.245 + 0.424i)21-s + (−0.193 − 0.334i)23-s + (−0.0999 − 0.173i)25-s − 0.517·27-s + (0.407 + 0.706i)29-s − 1.54·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.293263009\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.293263009\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.851 + 4.27i)T \) |
good | 3 | \( 1 + (0.232 - 0.402i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + (-2.23 - 3.87i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.30 - 2.26i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.927 + 1.60i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.19 - 3.80i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + (2.62 - 4.53i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.74 + 4.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.20 - 9.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.94 + 5.09i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.531 - 0.921i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.07 + 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.98 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.201 + 0.348i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.49 - 9.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.76 + 9.99i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + (-3.30 - 5.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.60 + 7.98i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.252127187054374828979871049347, −8.852824354849236426873007490109, −8.002216613133921115359248662212, −7.21701982001048536865678589706, −6.25604899549924526961073341691, −5.07441853359744808843249040946, −4.68684912860796024967609606181, −3.84213069198764951103072664370, −1.99061619739466079769792184893, −1.47428249780151644717969714494,
1.10426160358656272286730948281, 1.96100422276753790288348092539, 3.47057940230439593159554132123, 4.28089132861758014683263627656, 5.41736423859037669947254840168, 6.01416624990045349968931244003, 7.14670693084497251031215848763, 7.69359239744901148290029788019, 8.586725602987388193986081037225, 9.288276124818939796786893449975