L(s) = 1 | + (−1.19 − 0.755i)2-s + (0.859 + 1.80i)4-s − 3.16i·5-s − 7-s + (0.336 − 2.80i)8-s + (−2.39 + 3.78i)10-s − 5.44i·11-s − 3.61i·13-s + (1.19 + 0.755i)14-s + (−2.52 + 3.10i)16-s + 3.27·17-s − 3.20i·19-s + (5.72 − 2.72i)20-s + (−4.11 + 6.51i)22-s + 0.673·23-s + ⋯ |
L(s) = 1 | + (−0.845 − 0.534i)2-s + (0.429 + 0.903i)4-s − 1.41i·5-s − 0.377·7-s + (0.119 − 0.992i)8-s + (−0.757 + 1.19i)10-s − 1.64i·11-s − 1.00i·13-s + (0.319 + 0.201i)14-s + (−0.630 + 0.775i)16-s + 0.794·17-s − 0.735i·19-s + (1.28 − 0.608i)20-s + (−0.877 + 1.38i)22-s + 0.140·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8228761136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8228761136\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 + 0.755i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 3.16iT - 5T^{2} \) |
| 11 | \( 1 + 5.44iT - 11T^{2} \) |
| 13 | \( 1 + 3.61iT - 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 3.20iT - 19T^{2} \) |
| 23 | \( 1 - 0.673T + 23T^{2} \) |
| 29 | \( 1 - 2.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 - 11.9iT - 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 12.5iT - 43T^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 - 0.291iT - 53T^{2} \) |
| 59 | \( 1 - 0.0209iT - 59T^{2} \) |
| 61 | \( 1 - 5.34iT - 61T^{2} \) |
| 67 | \( 1 - 6.20iT - 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 1.35T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−8.767835824926251168121025121176, −8.605469694126197510610021046717, −7.87443927900162123206128599622, −6.74995646109256930700690348797, −5.71279903394829055622062815236, −4.90166338385537414235927119903, −3.58029154908434430032690687292, −2.90871610730689070070035501071, −1.25229572277432428014701730170, −0.47180545967596727826058612331,
1.73342674523395717640201322531, 2.63696419350305825905616727614, 3.92378892762383274644189578186, 5.10605961989742429429043896795, 6.29231934139978545662672693094, 6.71966324774165157227746170881, 7.44197422464055605646848530517, 8.046746603207701670185602003811, 9.378508842050627645291836054684, 9.779838595437894509468977524386