L(s) = 1 | − 5-s + 1.26·7-s − 2.26·11-s − 5.46·13-s − 0.732·17-s + 2.46·19-s + 3.46·23-s + 25-s + 7.19·29-s + 3·31-s − 1.26·35-s + 0.732·37-s − 3.19·41-s + 10.1·43-s − 5.26·47-s − 5.39·49-s − 3.26·53-s + 2.26·55-s − 11.7·59-s + 4·61-s + 5.46·65-s + 3.46·67-s − 0.267·71-s + 9.66·73-s − 2.87·77-s − 8.53·79-s − 8.19·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.479·7-s − 0.683·11-s − 1.51·13-s − 0.177·17-s + 0.565·19-s + 0.722·23-s + 0.200·25-s + 1.33·29-s + 0.538·31-s − 0.214·35-s + 0.120·37-s − 0.499·41-s + 1.55·43-s − 0.768·47-s − 0.770·49-s − 0.448·53-s + 0.305·55-s − 1.52·59-s + 0.512·61-s + 0.677·65-s + 0.423·67-s − 0.0317·71-s + 1.13·73-s − 0.327·77-s − 0.960·79-s − 0.899·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 + 0.267T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
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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
−7.80171414633887399742033438750, −7.04742228712044435066695211149, −6.38109401171130524449194759519, −5.21094763221445756911937585270, −4.92935864150635176242634353537, −4.15379975517218578533494015773, −2.98410888956672847031971331102, −2.50659840135976117015935080287, −1.23954087694080214688392628567, 0,
1.23954087694080214688392628567, 2.50659840135976117015935080287, 2.98410888956672847031971331102, 4.15379975517218578533494015773, 4.92935864150635176242634353537, 5.21094763221445756911937585270, 6.38109401171130524449194759519, 7.04742228712044435066695211149, 7.80171414633887399742033438750