| L(s) = 1 | − 5-s + 7-s − 3·9-s − 13-s + 6·17-s + 4·19-s + 25-s − 2·29-s − 8·31-s − 35-s + 2·37-s + 6·41-s − 4·43-s + 3·45-s − 8·47-s + 49-s − 10·53-s + 4·59-s + 14·61-s − 3·63-s + 65-s + 4·67-s + 12·71-s − 10·73-s + 8·79-s + 9·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 1.79·61-s − 0.377·63-s + 0.124·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s + 0.900·79-s + 81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560194604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560194604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.385771385812356079726347914926, −7.82348766905204698881999006311, −7.28116690475005017627912262363, −6.22639347846016491123713142429, −5.41549964373473641171043939100, −4.95094961367296816114150856302, −3.67662269296931945677806057545, −3.17872208219740989131030166308, −2.01975278282615221991503440065, −0.72793771617631361103693325069,
0.72793771617631361103693325069, 2.01975278282615221991503440065, 3.17872208219740989131030166308, 3.67662269296931945677806057545, 4.95094961367296816114150856302, 5.41549964373473641171043939100, 6.22639347846016491123713142429, 7.28116690475005017627912262363, 7.82348766905204698881999006311, 8.385771385812356079726347914926
