| L(s) = 1 | − 5-s + 4·7-s − 11-s + 2·13-s − 6·17-s − 4·19-s − 8·23-s + 25-s + 10·29-s − 4·35-s − 10·37-s + 6·41-s + 8·43-s + 9·49-s + 2·53-s + 55-s + 12·59-s − 2·61-s − 2·65-s + 12·67-s + 8·71-s − 2·73-s − 4·77-s + 8·79-s − 8·83-s + 6·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.85·29-s − 0.676·35-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 9/7·49-s + 0.274·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s + 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.961201214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.961201214\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.054470850673010220392765116729, −7.19473582723212329443299053031, −6.50249462656148603863329585364, −5.75430404210632195045390813428, −4.85092156831089606999094490025, −4.36334471271386614949825309738, −3.75144377870165057222742075279, −2.41607692132932969511501665388, −1.91933874623480413716766008018, −0.69102017863468173994172909292,
0.69102017863468173994172909292, 1.91933874623480413716766008018, 2.41607692132932969511501665388, 3.75144377870165057222742075279, 4.36334471271386614949825309738, 4.85092156831089606999094490025, 5.75430404210632195045390813428, 6.50249462656148603863329585364, 7.19473582723212329443299053031, 8.054470850673010220392765116729
