| L(s) = 1 | + 3-s + 3·5-s + 9-s − 11-s + 3·13-s + 3·15-s − 17-s + 6·19-s + 2·23-s + 4·25-s + 27-s − 6·29-s − 33-s − 3·37-s + 3·39-s + 11·43-s + 3·45-s − 51-s + 9·53-s − 3·55-s + 6·57-s + 4·59-s − 14·61-s + 9·65-s − 7·67-s + 2·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.417·23-s + 4/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.493·37-s + 0.480·39-s + 1.67·43-s + 0.447·45-s − 0.140·51-s + 1.23·53-s − 0.404·55-s + 0.794·57-s + 0.520·59-s − 1.79·61-s + 1.11·65-s − 0.855·67-s + 0.240·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.411776390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.411776390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.35532859874684, −13.12682163721433, −12.34442993370792, −11.99939537019039, −11.21208159069478, −10.86922790547478, −10.32702442029175, −9.908860682967351, −9.288188057101325, −9.120405398189059, −8.638629203132073, −7.943842184071360, −7.355512324956667, −7.115398531212753, −6.267706094998476, −5.852502176216183, −5.501404318227570, −4.902313838260678, −4.216445328507441, −3.602541304049623, −3.000733548225082, −2.541975435825337, −1.821005722764832, −1.406815668149471, −0.6499985546683725,
0.6499985546683725, 1.406815668149471, 1.821005722764832, 2.541975435825337, 3.000733548225082, 3.602541304049623, 4.216445328507441, 4.902313838260678, 5.501404318227570, 5.852502176216183, 6.267706094998476, 7.115398531212753, 7.355512324956667, 7.943842184071360, 8.638629203132073, 9.120405398189059, 9.288188057101325, 9.908860682967351, 10.32702442029175, 10.86922790547478, 11.21208159069478, 11.99939537019039, 12.34442993370792, 13.12682163721433, 13.35532859874684
