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
\(L(1)\) |
\(\approx\) |
\(2.193565406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.193565406\) |
\(L(\frac{3}{2})\) |
|
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
−9.049169778547849798081972117318, −8.627002534402446439384452238419, −7.41270262843343530154694316012, −6.13739376553276861538600949200, −5.84121314544139093765335124323, −4.69027784343016115771568144192, −4.10778940386051735014223669123, −3.07650483960987598229361905007, −1.76524561892432315484268807150, −0.62493901999726350076248227045,
2.34054952403822952307172964941, 2.88720069476816821133681898224, 4.00099447003491729689696995403, 4.82884419265733867747847259311, 5.90740223398885006846269145271, 6.72857631815366578522049016129, 7.15075878081515113124404007860, 7.87189931175719888572244989631, 8.973773390822871323190997608839, 10.12326594806568025646488299183