L(s) = 1 | + (−0.0388 − 0.0672i)2-s + (0.5 − 0.866i)3-s + (0.996 − 1.72i)4-s + (1.10 + 1.90i)5-s − 0.0776·6-s + (2.64 + 0.0672i)7-s − 0.310·8-s + (−0.499 − 0.866i)9-s + (0.0855 − 0.148i)10-s + (−1.45 + 2.52i)11-s + (−0.996 − 1.72i)12-s − 13-s + (−0.0981 − 0.180i)14-s + 2.20·15-s + (−1.98 − 3.43i)16-s + (0.196 − 0.341i)17-s + ⋯ |
L(s) = 1 | + (−0.0274 − 0.0475i)2-s + (0.288 − 0.499i)3-s + (0.498 − 0.863i)4-s + (0.492 + 0.852i)5-s − 0.0317·6-s + (0.999 + 0.0254i)7-s − 0.109·8-s + (−0.166 − 0.288i)9-s + (0.0270 − 0.0468i)10-s + (−0.439 + 0.761i)11-s + (−0.287 − 0.498i)12-s − 0.277·13-s + (−0.0262 − 0.0482i)14-s + 0.568·15-s + (−0.495 − 0.858i)16-s + (0.0477 − 0.0827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60017 - 0.505884i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60017 - 0.505884i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.64 - 0.0672i)T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.0388 + 0.0672i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.10 - 1.90i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.45 - 2.52i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.196 + 0.341i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0623 + 0.108i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.69 + 2.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.642T + 29T^{2} \) |
| 31 | \( 1 + (2.97 - 5.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.09 + 3.62i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.434T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + (-4.73 - 8.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.66 + 6.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.19 - 2.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.07 - 5.31i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + (-3.77 + 6.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.78T + 83T^{2} \) |
| 89 | \( 1 + (-3.20 - 5.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−11.70778431416009833719716714127, −10.71777447904852497400494264103, −10.19025855057495809140471430863, −9.013790963004416974365951101609, −7.67381886000814504562417125102, −6.92539499247113263524569323744, −5.90193486282536522176264429331, −4.76694391015261845560203288279, −2.67949049897543853959360811084, −1.73499768391214493231037139253,
1.98845905607278322101647626027, 3.49858640358184053307376660369, 4.78739935785386298098803501587, 5.77114270021148784098211248784, 7.37865487824049865355116545737, 8.286130296011721206751744728124, 8.871655961571537012566422768189, 10.08780784060744257945004004463, 11.20186547737496851652541002139, 11.85034844066598553038672625726