L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.959 + 0.281i)3-s + (−0.142 − 0.989i)4-s + (0.415 − 0.909i)6-s − 1.81i·7-s + (0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + 0.284·11-s + (0.415 + 0.909i)12-s + (1.37 + 1.19i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)18-s + 1.08i·19-s + (0.512 + 1.74i)21-s + (−0.186 + 0.215i)22-s + ⋯ |
L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.959 + 0.281i)3-s + (−0.142 − 0.989i)4-s + (0.415 − 0.909i)6-s − 1.81i·7-s + (0.841 + 0.540i)8-s + (0.841 − 0.540i)9-s + 0.284·11-s + (0.415 + 0.909i)12-s + (1.37 + 1.19i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)18-s + 1.08i·19-s + (0.512 + 1.74i)21-s + (−0.186 + 0.215i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4540255234\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4540255234\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.81iT - T^{2} \) |
| 11 | \( 1 - 0.284T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.08iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.81iT - T^{2} \) |
| 31 | \( 1 + 1.51iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.68T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.91T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.563iT - T^{2} \) |
| 97 | \( 1 + 0.284T + T^{2} \) |
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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
−9.566974651866522328012902883976, −8.055842989620565125322608142991, −7.69708109378563967813772693363, −6.78315708564438332244515900693, −6.22248138513001112385307602032, −5.41263286665319115717313654060, −4.26286794105143852689646343218, −3.94336142677950037292621403080, −1.67234156056442930718178238898, −0.48935273346476753254294368942,
1.47650705043448272081151416953, 2.40030282703516699244742858961, 3.43062018570299521503068427378, 4.82603252184221225210623853650, 5.36381234970157321451240062684, 6.46534286212368909642990301565, 7.07702154867784890425600563857, 8.193300695907305300828805979718, 8.822158703955563031513139736901, 9.476222254481463939866979989543