L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.41 + 2.44i)5-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−1.41 + 2.44i)10-s + (−0.5 + 0.866i)11-s − 4.24·13-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.44i)17-s + (−1.5 + 2.59i)18-s + (0.707 + 1.22i)19-s − 2.82·20-s − 0.999·22-s + (−3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.632 + 1.09i)5-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.447 + 0.774i)10-s + (−0.150 + 0.261i)11-s − 1.17·13-s + (−0.125 − 0.216i)16-s + (−0.342 + 0.594i)17-s + (−0.353 + 0.612i)18-s + (0.162 + 0.280i)19-s − 0.632·20-s − 0.213·22-s + (−0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.781419505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.781419505\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.707 - 1.22i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (3.53 + 6.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.07 - 12.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.12 + 3.67i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-4.24 + 7.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 + (-9.19 - 15.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−10.37363611956408009205348906307, −9.541848728533306257612743801610, −8.316917306575182513391884798859, −7.60422103181112350605046160850, −6.76810549761453412032274550872, −6.21045957179983763022317803790, −5.05547631760692165679752550972, −4.35008787844682506633222158031, −2.90798021682824209640675722723, −2.10007316785247457113001795570,
0.69090053827920293141263638921, 1.91466849033878411543357018002, 3.12278555001894664536278834516, 4.36969082957107006493974598273, 5.02742531725320234416667921299, 5.88749475774885026814310202252, 6.90414035134818985801014931290, 7.981334382265610009495601292494, 9.129899054122288020744107854881, 9.505563028476838408901493146265