L(s) = 1 | + (0.5 − 0.866i)3-s + (1 − 1.73i)4-s + (−1.27 − 2.21i)5-s + (−2.58 + 0.557i)7-s + (−0.499 − 0.866i)9-s + (−3.08 + 5.34i)11-s + (−0.999 − 1.73i)12-s − 2.06·13-s − 2.55·15-s + (−1.99 − 3.46i)16-s + (1 − 1.73i)17-s + (0.223 + 0.387i)19-s − 5.10·20-s + (−0.810 + 2.51i)21-s + (0.276 + 0.478i)23-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.570 − 0.988i)5-s + (−0.977 + 0.210i)7-s + (−0.166 − 0.288i)9-s + (−0.930 + 1.61i)11-s + (−0.288 − 0.499i)12-s − 0.573·13-s − 0.659·15-s + (−0.499 − 0.866i)16-s + (0.242 − 0.420i)17-s + (0.0513 + 0.0889i)19-s − 1.14·20-s + (−0.176 + 0.549i)21-s + (0.0575 + 0.0997i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.113545 + 0.599217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113545 + 0.599217i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.58 - 0.557i)T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.27 + 2.21i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.08 - 5.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.223 - 0.387i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.276 - 0.478i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 + (-3.86 + 6.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + (-4.15 - 7.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 3.58i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.06 - 7.04i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.87 + 6.71i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0338 + 0.0585i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 + (-5.31 + 9.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.39 + 14.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + (6.15 + 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.752187229112758938475558271074, −9.009978576623732823278377762245, −7.75131467378251837107452803822, −7.27462654846059593715739391250, −6.26190652506586346646729841884, −5.24301923428664390608309679583, −4.46423796734062632391566853913, −2.86702189231908063230868945898, −1.84968371553826701174054429451, −0.25928849703570603498450021626,
2.69867613248757695729624363587, 3.22530981891750119989669103766, 3.87171521844227780051834535057, 5.44849364329614145303879181172, 6.55239140643983952990202182807, 7.24800684450683995963243409825, 8.100736962189396230938994952512, 8.769590710550955035297395375343, 10.01193484219829211232624578265, 10.69081476331115395541212064811