| L(s) = 1 | − 7-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 10·29-s + 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 4·59-s + 10·61-s − 4·67-s − 16·71-s + 14·73-s − 4·77-s − 8·79-s − 4·83-s − 10·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.85·29-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.488·67-s − 1.89·71-s + 1.63·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s − 1.05·89-s + 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.20140738474926, −13.36168266320457, −13.05538045829455, −12.39026551873054, −12.24482182098011, −11.45029751818188, −11.14316776841009, −10.61071252554244, −9.936068568753830, −9.414541891212222, −9.264430029142941, −8.550441908827513, −8.009434116593504, −7.348159395469007, −7.017888035127362, −6.409569665927279, −5.740011230520406, −5.577741524062822, −4.529899422832875, −4.132751049575124, −3.706012147320178, −2.900300581814198, −2.321151511088108, −1.617340792687209, −0.8788644280086566, 0,
0.8788644280086566, 1.617340792687209, 2.321151511088108, 2.900300581814198, 3.706012147320178, 4.132751049575124, 4.529899422832875, 5.577741524062822, 5.740011230520406, 6.409569665927279, 7.017888035127362, 7.348159395469007, 8.009434116593504, 8.550441908827513, 9.264430029142941, 9.414541891212222, 9.936068568753830, 10.61071252554244, 11.14316776841009, 11.45029751818188, 12.24482182098011, 12.39026551873054, 13.05538045829455, 13.36168266320457, 14.20140738474926
