| L(s) = 1 | + 2·3-s − 7-s + 9-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s − 2·21-s − 23-s − 4·27-s − 6·29-s − 6·31-s − 8·33-s + 6·37-s + 4·39-s − 6·41-s − 2·43-s + 12·47-s + 49-s + 12·51-s − 2·53-s + 8·57-s − 4·59-s − 6·61-s − 63-s + 10·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s − 0.769·27-s − 1.11·29-s − 1.07·31-s − 1.39·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s + 1.05·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s + 1.22·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.02647748700617, −12.83192949871151, −12.20607794650458, −11.72971292975090, −11.10275815700786, −10.76735749647012, −10.12630903648370, −9.700220186389258, −9.413316265318711, −8.850120337683000, −8.343437615755326, −7.902356540822542, −7.501944984524254, −7.260153885099055, −6.424856018514223, −5.636044047681441, −5.603781720089788, −5.005224635375075, −4.049949720893340, −3.740139190230942, −3.083284307928060, −2.926324563527023, −2.166826398474116, −1.622723207083497, −0.8436105343474921, 0,
0.8436105343474921, 1.622723207083497, 2.166826398474116, 2.926324563527023, 3.083284307928060, 3.740139190230942, 4.049949720893340, 5.005224635375075, 5.603781720089788, 5.636044047681441, 6.424856018514223, 7.260153885099055, 7.501944984524254, 7.902356540822542, 8.343437615755326, 8.850120337683000, 9.413316265318711, 9.700220186389258, 10.12630903648370, 10.76735749647012, 11.10275815700786, 11.72971292975090, 12.20607794650458, 12.83192949871151, 13.02647748700617
