L(s) = 1 | − 2.65·2-s + 5.04·4-s − 2.16·5-s − 2.02·7-s − 8.08·8-s + 5.74·10-s − 11-s + 0.318·13-s + 5.36·14-s + 11.3·16-s + 1.66·17-s + 0.0856·19-s − 10.9·20-s + 2.65·22-s + 6.29·23-s − 0.318·25-s − 0.844·26-s − 10.2·28-s + 2.14·29-s − 4.26·31-s − 14.0·32-s − 4.42·34-s + 4.37·35-s − 8.58·37-s − 0.227·38-s + 17.4·40-s − 3.06·41-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 2.52·4-s − 0.967·5-s − 0.764·7-s − 2.85·8-s + 1.81·10-s − 0.301·11-s + 0.0882·13-s + 1.43·14-s + 2.84·16-s + 0.404·17-s + 0.0196·19-s − 2.44·20-s + 0.565·22-s + 1.31·23-s − 0.0636·25-s − 0.165·26-s − 1.92·28-s + 0.397·29-s − 0.765·31-s − 2.47·32-s − 0.759·34-s + 0.739·35-s − 1.41·37-s − 0.0368·38-s + 2.76·40-s − 0.478·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 13 | \( 1 - 0.318T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 - 0.0856T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 - 4.56T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 4.06T + 71T^{2} \) |
| 73 | \( 1 - 5.81T + 73T^{2} \) |
| 79 | \( 1 - 7.59T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.78552839829959838541298776356, −7.28729828315687867877985554651, −6.71649465676511287271536201282, −5.95196277868269798163360831677, −4.93621671156177817197669164196, −3.59357742677772896369291389171, −3.09115206204992587251981888941, −2.04804898273480102548309523599, −0.904564921388826744152560727779, 0,
0.904564921388826744152560727779, 2.04804898273480102548309523599, 3.09115206204992587251981888941, 3.59357742677772896369291389171, 4.93621671156177817197669164196, 5.95196277868269798163360831677, 6.71649465676511287271536201282, 7.28729828315687867877985554651, 7.78552839829959838541298776356