| L(s) = 1 | + 3·5-s + 4·7-s − 3·9-s + 3·13-s + 3·17-s + 4·19-s − 8·23-s + 4·25-s − 5·29-s − 4·31-s + 12·35-s + 11·37-s + 7·41-s − 12·43-s − 9·45-s − 8·47-s + 9·49-s − 53-s − 4·59-s − 2·61-s − 12·63-s + 9·65-s + 4·67-s + 12·71-s + 10·73-s + 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 9-s + 0.832·13-s + 0.727·17-s + 0.917·19-s − 1.66·23-s + 4/5·25-s − 0.928·29-s − 0.718·31-s + 2.02·35-s + 1.80·37-s + 1.09·41-s − 1.82·43-s − 1.34·45-s − 1.16·47-s + 9/7·49-s − 0.137·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s + 1.11·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.289811341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289811341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−9.871100332584929424091507406903, −9.315729680767934174365366461202, −8.194968820725900543272135034508, −7.83217233414190971770307310117, −6.30681177281106592088147922223, −5.65688575675512458890240062698, −5.05440153409789039731500985582, −3.66665168393547624429364752240, −2.29728473273749431649763829937, −1.40572175942725518657269326631,
1.40572175942725518657269326631, 2.29728473273749431649763829937, 3.66665168393547624429364752240, 5.05440153409789039731500985582, 5.65688575675512458890240062698, 6.30681177281106592088147922223, 7.83217233414190971770307310117, 8.194968820725900543272135034508, 9.315729680767934174365366461202, 9.871100332584929424091507406903
