| L(s) = 1 | − 5-s − 7-s − 2·11-s − 17-s + 4·19-s + 4·23-s − 4·25-s + 8·29-s − 4·31-s + 35-s − 7·37-s + 5·41-s − 43-s + 9·47-s + 49-s − 2·53-s + 2·55-s − 3·59-s − 2·61-s + 4·67-s − 12·71-s − 16·73-s + 2·77-s + 15·79-s + 3·83-s + 85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.242·17-s + 0.917·19-s + 0.834·23-s − 4/5·25-s + 1.48·29-s − 0.718·31-s + 0.169·35-s − 1.15·37-s + 0.780·41-s − 0.152·43-s + 1.31·47-s + 1/7·49-s − 0.274·53-s + 0.269·55-s − 0.390·59-s − 0.256·61-s + 0.488·67-s − 1.42·71-s − 1.87·73-s + 0.227·77-s + 1.68·79-s + 0.329·83-s + 0.108·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.58794636549087719079075710681, −7.18617274624729488372235725433, −6.31767339512495890710643917479, −5.53771768156548149184613512389, −4.85786785681312537691792387582, −4.00461938574110629761585323471, −3.19471728756461723415387303444, −2.48809879300485040829241569901, −1.22003559648175783296035360833, 0,
1.22003559648175783296035360833, 2.48809879300485040829241569901, 3.19471728756461723415387303444, 4.00461938574110629761585323471, 4.85786785681312537691792387582, 5.53771768156548149184613512389, 6.31767339512495890710643917479, 7.18617274624729488372235725433, 7.58794636549087719079075710681
