L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.841 − 0.540i)3-s + (−0.959 + 0.281i)4-s + (0.415 − 0.909i)6-s − 1.51i·7-s + (−0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s − 1.91·11-s + (0.959 + 0.281i)12-s + (1.49 − 0.215i)14-s + (0.841 − 0.540i)16-s + (−0.841 + 0.540i)18-s + 1.81i·19-s + (−0.817 + 1.27i)21-s + (−0.273 − 1.89i)22-s + ⋯ |
L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.841 − 0.540i)3-s + (−0.959 + 0.281i)4-s + (0.415 − 0.909i)6-s − 1.51i·7-s + (−0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s − 1.91·11-s + (0.959 + 0.281i)12-s + (1.49 − 0.215i)14-s + (0.841 − 0.540i)16-s + (−0.841 + 0.540i)18-s + 1.81i·19-s + (−0.817 + 1.27i)21-s + (−0.273 − 1.89i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2709976748\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2709976748\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.51iT - T^{2} \) |
| 11 | \( 1 + 1.91T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.81iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.51iT - T^{2} \) |
| 31 | \( 1 - 1.97iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 0.830T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.68T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.08iT - T^{2} \) |
| 97 | \( 1 + 1.91T + T^{2} \) |
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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
−9.915411180569273505460833811433, −8.496527941511592310680743488880, −7.73924828491116710202652371896, −7.43044524173075439652907654003, −6.65292335070755628596519847894, −5.70201818103537744369422476957, −5.14675861266627290984569925382, −4.28388387351064876737646817396, −3.25411460284426448719575499226, −1.41114859817826356852797458790,
0.21067066427896650737313022323, 2.31496234839932843104106855748, 2.76933209393718318172506862138, 4.15926447630209082871961429122, 4.95688353425821818503310816831, 5.59797369390279907549930120361, 6.10199032950348793546427806738, 7.62762734096181417469619864582, 8.443232273740814134533040307696, 9.429498249640891416349232603207