| 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
−7.56591841669389549344613539460, −7.20577037524701959547829928381, −6.14047182669972384291407091622, −5.64030682883616401782196055611, −4.87613728415884360324154040714, −4.10809580582249546972829643047, −3.15770146059131870613415798389, −2.25339717367876448610919124359, −1.46289063938847208106552932535, 0,
1.46289063938847208106552932535, 2.25339717367876448610919124359, 3.15770146059131870613415798389, 4.10809580582249546972829643047, 4.87613728415884360324154040714, 5.64030682883616401782196055611, 6.14047182669972384291407091622, 7.20577037524701959547829928381, 7.56591841669389549344613539460
