| L(s) = 1 | + 5-s + 11-s + 7·19-s + 6·23-s + 25-s + 7·29-s − 31-s − 2·37-s − 9·41-s + 6·43-s − 2·47-s − 7·49-s + 55-s + 3·59-s − 10·61-s + 2·67-s + 71-s − 4·79-s − 6·83-s + 7·89-s + 7·95-s + 2·97-s + 9·101-s + 6·103-s − 2·107-s + 3·109-s + 6·115-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s + 1.60·19-s + 1.25·23-s + 1/5·25-s + 1.29·29-s − 0.179·31-s − 0.328·37-s − 1.40·41-s + 0.914·43-s − 0.291·47-s − 49-s + 0.134·55-s + 0.390·59-s − 1.28·61-s + 0.244·67-s + 0.118·71-s − 0.450·79-s − 0.658·83-s + 0.741·89-s + 0.718·95-s + 0.203·97-s + 0.895·101-s + 0.591·103-s − 0.193·107-s + 0.287·109-s + 0.559·115-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
| \(L(1)\) |
\(\approx\) |
\(2.483846630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483846630\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.013782739305065209674938865513, −7.20563270867233099742134342004, −6.68884312447935055180410682460, −5.87192668984368112346881578133, −5.13384207775117203035395052395, −4.58034932210475949768272897919, −3.39601205279103558946922706172, −2.90705015777726251765995675022, −1.72754772230830419362082532352, −0.860916160926714767313031169310,
0.860916160926714767313031169310, 1.72754772230830419362082532352, 2.90705015777726251765995675022, 3.39601205279103558946922706172, 4.58034932210475949768272897919, 5.13384207775117203035395052395, 5.87192668984368112346881578133, 6.68884312447935055180410682460, 7.20563270867233099742134342004, 8.013782739305065209674938865513
