| L(s) = 1 | − 2·3-s − 7-s + 9-s + 2·13-s − 5·17-s − 8·19-s + 2·21-s + 23-s + 4·27-s − 5·29-s + 5·31-s + 7·37-s − 4·39-s − 7·41-s − 4·43-s + 2·47-s − 6·49-s + 10·51-s − 53-s + 16·57-s − 3·59-s − 6·61-s − 63-s − 13·67-s − 2·69-s − 13·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.21·17-s − 1.83·19-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 0.928·29-s + 0.898·31-s + 1.15·37-s − 0.640·39-s − 1.09·41-s − 0.609·43-s + 0.291·47-s − 6/7·49-s + 1.40·51-s − 0.137·53-s + 2.11·57-s − 0.390·59-s − 0.768·61-s − 0.125·63-s − 1.58·67-s − 0.240·69-s − 1.54·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5157800717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5157800717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.61740417591962582901194978138, −6.71224977744596447447526051161, −6.27201911569989834565020806029, −5.94483975293851586416450649157, −4.83897941359198417242241098670, −4.48912079134868590226289854822, −3.55020406993413560665424356207, −2.56482427318513697789181233058, −1.62500030556551833895154256831, −0.36203022302196198863017700598,
0.36203022302196198863017700598, 1.62500030556551833895154256831, 2.56482427318513697789181233058, 3.55020406993413560665424356207, 4.48912079134868590226289854822, 4.83897941359198417242241098670, 5.94483975293851586416450649157, 6.27201911569989834565020806029, 6.71224977744596447447526051161, 7.61740417591962582901194978138
