L(s) = 1 | − 5-s + 4·7-s − 3·9-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s − 6·37-s − 6·41-s − 8·43-s + 3·45-s − 4·47-s + 9·49-s − 6·53-s − 4·55-s − 4·59-s + 2·61-s − 12·63-s − 2·65-s + 8·67-s − 6·73-s + 16·77-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s − 1.51·63-s − 0.248·65-s + 0.977·67-s − 0.702·73-s + 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.427581995\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427581995\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.75069621452783403309599418360, −10.98792460132555645103756947006, −9.759879680318004905765589416029, −8.438761280447847026288494531448, −8.187189248591461729679142296814, −6.81982863783073968912914119025, −5.61529786752672156928392151688, −4.55607707806255331316875895933, −3.31415635628336508949565358363, −1.46465509347525672726427035614,
1.46465509347525672726427035614, 3.31415635628336508949565358363, 4.55607707806255331316875895933, 5.61529786752672156928392151688, 6.81982863783073968912914119025, 8.187189248591461729679142296814, 8.438761280447847026288494531448, 9.759879680318004905765589416029, 10.98792460132555645103756947006, 11.75069621452783403309599418360