L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s − 0.999·6-s + (−2.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)10-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)12-s − 6·13-s + (2 + 1.73i)14-s + 3·15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s − 0.408·6-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (−0.150 + 0.261i)11-s + (0.144 + 0.249i)12-s − 1.66·13-s + (0.534 + 0.462i)14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.343477 + 0.451495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.343477 + 0.451495i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13T + 83T^{2} \) |
| 89 | \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−10.05589631496281501368192555717, −9.676213546611871581085748678867, −8.870372306342280414195622581316, −7.48257196277459954240382371188, −7.14320863162329243498013726139, −6.15555771762743253557862360701, −5.09353705955192793856188714465, −3.42561085167064956658946670582, −2.76383080759142530179728279853, −1.92237527958281162874417283040,
0.26067682897197828543551700295, 2.04828039071573666403744488370, 3.50795430069871278115690057824, 4.76075201385361770602888148257, 5.34602186115786228687311210255, 6.30777994158210965966384176577, 7.37049190507704357889919229485, 8.150304588394075340463508576525, 9.270344512506887448638965920088, 9.487620842230493111988887035186