| L(s) = 1 | − 5-s + 3·7-s + 2·11-s − 2·13-s + 4·17-s + 8·19-s − 3·23-s + 25-s − 29-s − 3·35-s − 4·37-s + 5·41-s + 8·43-s − 7·47-s + 2·49-s − 2·53-s − 2·55-s + 14·59-s + 7·61-s + 2·65-s + 3·67-s − 2·71-s + 4·73-s + 6·77-s + 6·79-s − 9·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.185·29-s − 0.507·35-s − 0.657·37-s + 0.780·41-s + 1.21·43-s − 1.02·47-s + 2/7·49-s − 0.274·53-s − 0.269·55-s + 1.82·59-s + 0.896·61-s + 0.248·65-s + 0.366·67-s − 0.237·71-s + 0.468·73-s + 0.683·77-s + 0.675·79-s − 0.987·83-s − 0.433·85-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
| \(L(1)\) |
\(\approx\) |
\(2.393778280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393778280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.990921188090353684329255048335, −7.40207927874331357562089818268, −6.82133651842454103936172110053, −5.61069069850025403436598361067, −5.28490659558466783587734238259, −4.37216752275413301188092512055, −3.68672752112726026926723145459, −2.80122069471657119570899805064, −1.69089801741608264928632165843, −0.857470382493630375229737718449,
0.857470382493630375229737718449, 1.69089801741608264928632165843, 2.80122069471657119570899805064, 3.68672752112726026926723145459, 4.37216752275413301188092512055, 5.28490659558466783587734238259, 5.61069069850025403436598361067, 6.82133651842454103936172110053, 7.40207927874331357562089818268, 7.990921188090353684329255048335
