| L(s) = 1 | + 5-s − 7-s + 5·11-s + 13-s − 3·17-s − 6·19-s − 7·23-s + 25-s + 6·29-s − 10·31-s − 35-s − 11·37-s − 5·41-s − 4·43-s + 2·47-s − 6·49-s + 5·53-s + 5·55-s + 61-s + 65-s − 4·67-s + 5·71-s + 6·73-s − 5·77-s + 11·79-s + 8·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.50·11-s + 0.277·13-s − 0.727·17-s − 1.37·19-s − 1.45·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s − 0.169·35-s − 1.80·37-s − 0.780·41-s − 0.609·43-s + 0.291·47-s − 6/7·49-s + 0.686·53-s + 0.674·55-s + 0.128·61-s + 0.124·65-s − 0.488·67-s + 0.593·71-s + 0.702·73-s − 0.569·77-s + 1.23·79-s + 0.878·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.116433095714925874554481810538, −6.84949985639853006626454903408, −6.61929202163028444338927041194, −5.92386863280434015415341649435, −4.97612783770889451149779102206, −4.01528374154217637934876702609, −3.55231506443078593567727205719, −2.21501177855221154688146802232, −1.56235757553865713704418134549, 0,
1.56235757553865713704418134549, 2.21501177855221154688146802232, 3.55231506443078593567727205719, 4.01528374154217637934876702609, 4.97612783770889451149779102206, 5.92386863280434015415341649435, 6.61929202163028444338927041194, 6.84949985639853006626454903408, 8.116433095714925874554481810538
