L(s) = 1 | + (1 + 1.41i)3-s + (−1.00 + 2.82i)9-s − 6·11-s + 5.65i·17-s + 8.48i·19-s + 5·25-s + (−5.00 + 1.41i)27-s + (−6 − 8.48i)33-s − 11.3i·41-s + 8.48i·43-s + 7·49-s + (−8.00 + 5.65i)51-s + (−12 + 8.48i)57-s + 6·59-s − 8.48i·67-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s + (−0.333 + 0.942i)9-s − 1.80·11-s + 1.37i·17-s + 1.94i·19-s + 25-s + (−0.962 + 0.272i)27-s + (−1.04 − 1.47i)33-s − 1.76i·41-s + 1.29i·43-s + 49-s + (−1.12 + 0.792i)51-s + (−1.58 + 1.12i)57-s + 0.781·59-s − 1.03i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.622150 + 1.20190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.622150 + 1.20190i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 8.48iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.34978052077095511992188404187, −10.08468491543284455159461073244, −8.790302036599455494425237302694, −8.162167699761703250175291054002, −7.49955535214414209473593430205, −5.95630682999826388559510429731, −5.23749201718865987480865849146, −4.13726076763144534312633150309, −3.19057485211952431031773564650, −2.04263476431434176938798796144,
0.60959836316426054745840675418, 2.49289565774882518970617759263, 2.96179654937690131647998420245, 4.69186723337989744382247992323, 5.52234145276171359986549740330, 6.89652540821914029109718029367, 7.31850640523112292206964249084, 8.301570930389600637753947718860, 9.014021864945821662185383300290, 9.926124375736567438467452817323