L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 6·13-s + 16-s + 2·17-s − 4·19-s − 22-s − 6·26-s + 10·29-s + 32-s + 2·34-s − 6·37-s − 4·38-s − 2·41-s − 4·43-s − 44-s − 8·47-s − 7·49-s − 6·52-s − 10·53-s + 10·58-s + 4·59-s − 2·61-s + 64-s + 4·67-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.213·22-s − 1.17·26-s + 1.85·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 1.16·47-s − 49-s − 0.832·52-s − 1.37·53-s + 1.31·58-s + 0.520·59-s − 0.256·61-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.953519059448833265921148977490, −6.87516314576116310072723973234, −6.61930482181016241055363584086, −5.50519254084154730512895831306, −4.91176036475039380235487105270, −4.34547747847136462528881406880, −3.22266685034392941829856826252, −2.59758594639330277923535938645, −1.61525671840520505272701261819, 0,
1.61525671840520505272701261819, 2.59758594639330277923535938645, 3.22266685034392941829856826252, 4.34547747847136462528881406880, 4.91176036475039380235487105270, 5.50519254084154730512895831306, 6.61930482181016241055363584086, 6.87516314576116310072723973234, 7.953519059448833265921148977490