L(s) = 1 | + (1.5 + 0.866i)3-s + 5-s + (2.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + 3·11-s + (−0.5 − 0.866i)13-s + (1.5 + 0.866i)15-s + (−1.5 − 2.59i)17-s + (2.5 − 4.33i)19-s + (3 + 3.46i)21-s − 23-s − 4·25-s + 5.19i·27-s + (−4.5 + 7.79i)29-s + (2 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + 0.447·5-s + (0.944 + 0.327i)7-s + (0.5 + 0.866i)9-s + 0.904·11-s + (−0.138 − 0.240i)13-s + (0.387 + 0.223i)15-s + (−0.363 − 0.630i)17-s + (0.573 − 0.993i)19-s + (0.654 + 0.755i)21-s − 0.208·23-s − 0.800·25-s + 0.999i·27-s + (−0.835 + 1.44i)29-s + (0.359 − 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.661830726\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.661830726\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.5 - 11.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.779234120622260103825673264826, −9.217845918084275416428514865098, −8.572924509691368573195944604944, −7.64715858611448764716170142053, −6.84199170077041713333442427253, −5.49046248812668576999836412958, −4.79243623790019665119544464351, −3.76790532616847037004619472395, −2.60927833244559101138015319216, −1.59898483877940763721174448712,
1.39928190328830203230809695440, 2.13329657963466697794068030030, 3.63520668692261658897688127217, 4.35161592462145940197429539428, 5.74309446452027759966945853436, 6.56404412218236368100298542352, 7.58504368879306099563759295735, 8.118943662606817609951073246647, 9.040151919805813430038464962107, 9.718064825195812181491392346196