| L(s) = 1 | + 3-s − 3·7-s + 9-s + 6·11-s − 2·13-s − 3·17-s + 6·19-s − 3·21-s + 23-s + 27-s + 9·29-s − 3·31-s + 6·33-s + 3·37-s − 2·39-s − 3·41-s − 4·47-s + 2·49-s − 3·51-s − 9·53-s + 6·57-s + 3·59-s + 8·61-s − 3·63-s + 3·67-s + 69-s − 9·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.727·17-s + 1.37·19-s − 0.654·21-s + 0.208·23-s + 0.192·27-s + 1.67·29-s − 0.538·31-s + 1.04·33-s + 0.493·37-s − 0.320·39-s − 0.468·41-s − 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.794·57-s + 0.390·59-s + 1.02·61-s − 0.377·63-s + 0.366·67-s + 0.120·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.75611434487720, −13.59823953144286, −12.94741444655662, −12.35623121706458, −12.12041323054252, −11.48331620717214, −11.10222530744034, −10.27283144050532, −9.759170700056558, −9.489977052213282, −9.156179811770339, −8.460922145811587, −8.107381776815156, −7.138045289394032, −6.948839961511736, −6.531745800432245, −5.933841993427247, −5.255357943890424, −4.468665867741233, −4.148670512727529, −3.318271964844962, −3.108182069893816, −2.403109345657348, −1.520346698558282, −1.003865674779443, 0,
1.003865674779443, 1.520346698558282, 2.403109345657348, 3.108182069893816, 3.318271964844962, 4.148670512727529, 4.468665867741233, 5.255357943890424, 5.933841993427247, 6.531745800432245, 6.948839961511736, 7.138045289394032, 8.107381776815156, 8.460922145811587, 9.156179811770339, 9.489977052213282, 9.759170700056558, 10.27283144050532, 11.10222530744034, 11.48331620717214, 12.12041323054252, 12.35623121706458, 12.94741444655662, 13.59823953144286, 13.75611434487720
