| L(s) = 1 | + 3·5-s + 4·7-s − 13-s + 3·17-s + 4·19-s + 4·25-s − 9·29-s + 4·31-s + 12·35-s − 37-s − 6·41-s − 8·43-s − 12·47-s + 9·49-s + 6·53-s − 61-s − 3·65-s + 4·67-s − 12·71-s + 11·73-s + 16·79-s − 12·83-s + 9·85-s + 3·89-s − 4·91-s + 12·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 0.277·13-s + 0.727·17-s + 0.917·19-s + 4/5·25-s − 1.67·29-s + 0.718·31-s + 2.02·35-s − 0.164·37-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s − 0.128·61-s − 0.372·65-s + 0.488·67-s − 1.42·71-s + 1.28·73-s + 1.80·79-s − 1.31·83-s + 0.976·85-s + 0.317·89-s − 0.419·91-s + 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.537710062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.537710062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−9.799999281303203285386475514980, −8.878539655843769294773456314485, −8.033910030135530589399167933473, −7.29957842837664580142557578027, −6.20376328566111560075210542997, −5.29484236286034362173688893149, −4.91673812495623205848110187798, −3.45423455586722913101065597862, −2.12069515781404083792870585763, −1.39451507835808804252991831473,
1.39451507835808804252991831473, 2.12069515781404083792870585763, 3.45423455586722913101065597862, 4.91673812495623205848110187798, 5.29484236286034362173688893149, 6.20376328566111560075210542997, 7.29957842837664580142557578027, 8.033910030135530589399167933473, 8.878539655843769294773456314485, 9.799999281303203285386475514980
