L(s) = 1 | + (1 + 1.73i)5-s + (1.5 + 2.59i)9-s + (−2 + 3.46i)11-s − 2·13-s + (−3 + 5.19i)17-s + (−4 − 6.92i)19-s + (0.500 − 0.866i)25-s + 6·29-s + (−4 + 6.92i)31-s + (1 + 1.73i)37-s − 2·41-s + 4·43-s + (−3 + 5.19i)45-s + (4 + 6.92i)47-s + (−3 + 5.19i)53-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s − 0.554·13-s + (−0.727 + 1.26i)17-s + (−0.917 − 1.58i)19-s + (0.100 − 0.173i)25-s + 1.11·29-s + (−0.718 + 1.24i)31-s + (0.164 + 0.284i)37-s − 0.312·41-s + 0.609·43-s + (−0.447 + 0.774i)45-s + (0.583 + 1.01i)47-s + (−0.412 + 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.800540 + 1.05229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.800540 + 1.05229i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.75119470221633556636948128829, −9.888895426491069983272782602191, −8.877228451552243776462101443328, −7.87452900486457864826104507420, −6.98325129076072899329028696731, −6.40896181237153161910598432055, −4.99955155236562778776697661570, −4.39536815587148840809396507316, −2.71034590778072371638674923356, −1.99529843829270034797447060986,
0.64074531005130499193893225227, 2.19851134550428847796667157282, 3.57036606451058228395370250296, 4.66172332010624618152436203994, 5.61257489309429871597531707200, 6.42724374279470323287100445823, 7.51204657321655526559470134300, 8.480634415049145512604342918048, 9.206956048839568331853901495115, 9.924390313348500875756118113815