| L(s) = 1 | + 4·4-s + 2·7-s − 13-s + 16·16-s + 26·19-s + 8·28-s + 59·31-s − 73·37-s + 83·43-s − 45·49-s − 4·52-s + 74·61-s + 64·64-s − 109·67-s + 143·73-s + 104·76-s + 11·79-s − 2·91-s + 2·97-s − 157·103-s + 71·109-s + 32·112-s + ⋯ |
| L(s) = 1 | + 4-s + 2/7·7-s − 0.0769·13-s + 16-s + 1.36·19-s + 2/7·28-s + 1.90·31-s − 1.97·37-s + 1.93·43-s − 0.918·49-s − 0.0769·52-s + 1.21·61-s + 64-s − 1.62·67-s + 1.95·73-s + 1.36·76-s + 0.139·79-s − 0.0219·91-s + 2/97·97-s − 1.52·103-s + 0.651·109-s + 2/7·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.543464629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543464629\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 2 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 59 T + p^{2} T^{2} \) |
| 37 | \( 1 + 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 83 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 74 T + p^{2} T^{2} \) |
| 67 | \( 1 + 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 143 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 2 T + p^{2} T^{2} \) |
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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
−10.37400386561098389546113341484, −9.580894774378702140322054357764, −8.418065307325777388638672936678, −7.58751752208981159556153892583, −6.82826985421381725085257541908, −5.86725549897338164405160800941, −4.90976385243019413285507089201, −3.50537046717136078797369863261, −2.47420040991955569893721416863, −1.17241441856296639148055129459,
1.17241441856296639148055129459, 2.47420040991955569893721416863, 3.50537046717136078797369863261, 4.90976385243019413285507089201, 5.86725549897338164405160800941, 6.82826985421381725085257541908, 7.58751752208981159556153892583, 8.418065307325777388638672936678, 9.580894774378702140322054357764, 10.37400386561098389546113341484
