| L(s) = 1 | + (0.142 − 0.989i)2-s + (−1.10 − 2.42i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−2.55 + 0.751i)6-s + (−0.986 − 0.634i)7-s + (−0.415 + 0.909i)8-s + (−2.69 + 3.11i)9-s + (−0.841 + 0.540i)10-s + (0.195 + 1.36i)11-s + (0.379 + 2.64i)12-s + (1.43 − 0.919i)13-s + (−0.768 + 0.886i)14-s + (−1.10 + 2.42i)15-s + (0.841 + 0.540i)16-s + (3.87 − 1.13i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.639 − 1.40i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−1.04 + 0.306i)6-s + (−0.372 − 0.239i)7-s + (−0.146 + 0.321i)8-s + (−0.898 + 1.03i)9-s + (−0.266 + 0.170i)10-s + (0.0590 + 0.410i)11-s + (0.109 + 0.762i)12-s + (0.396 − 0.254i)13-s + (−0.205 + 0.236i)14-s + (−0.286 + 0.626i)15-s + (0.210 + 0.135i)16-s + (0.939 − 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.137428 + 0.712412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137428 + 0.712412i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.63 + 4.00i)T \) |
| good | 3 | \( 1 + (1.10 + 2.42i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (0.986 + 0.634i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.195 - 1.36i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 0.919i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.87 + 1.13i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (5.55 + 1.62i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (3.94 - 1.15i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.74 + 3.81i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.83 + 3.27i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.48 - 5.17i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.90 + 10.7i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + (1.21 + 0.783i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-9.37 + 6.02i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.22 - 2.68i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.299 - 2.08i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.947 - 6.59i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.38 - 1.87i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.13 + 4.58i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (6.62 - 7.64i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.01 - 8.78i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (11.4 + 13.2i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.87216897849174365598764410457, −11.00416466393493861086292786853, −9.954248801096723810024692356540, −8.605922933970808797650103656046, −7.60480322388852641922096288774, −6.55929496130833111356759547713, −5.50444647450810656281604568167, −4.00782896678277049607580479817, −2.20927758210515940556599077296, −0.63206353484785635202710225428,
3.46362430189321236114008780500, 4.33997392086510087445565422988, 5.64095242526957298516358981329, 6.30732189079733518719713799301, 7.81872642145345271904207791626, 8.954677757702562348027990124640, 9.879729848321329277016286013841, 10.70422408843892649521692405949, 11.61250104044549901246817587327, 12.66947047054284599084620891049
