| L(s) = 1 | − 2·4-s − 4·5-s + 2·7-s − 3·9-s − 6·11-s + 4·13-s + 4·16-s − 6·19-s + 8·20-s + 6·23-s + 11·25-s − 4·28-s − 4·29-s − 6·31-s − 8·35-s + 6·36-s − 8·37-s + 3·41-s + 7·43-s + 12·44-s + 12·45-s + 47-s − 3·49-s − 8·52-s − 9·53-s + 24·55-s − 2·59-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 0.755·7-s − 9-s − 1.80·11-s + 1.10·13-s + 16-s − 1.37·19-s + 1.78·20-s + 1.25·23-s + 11/5·25-s − 0.755·28-s − 0.742·29-s − 1.07·31-s − 1.35·35-s + 36-s − 1.31·37-s + 0.468·41-s + 1.06·43-s + 1.80·44-s + 1.78·45-s + 0.145·47-s − 3/7·49-s − 1.10·52-s − 1.23·53-s + 3.23·55-s − 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 163 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−12.48626235146583554714277672207, −10.99632573201437702550357952641, −10.92315118072156934307382996948, −8.786931194290833006559128038298, −8.324796826494764068551514324559, −7.52091068808960772289628326952, −5.49628077301005674781360028304, −4.48576209360325789297312885526, −3.29187657679565605273397221600, 0,
3.29187657679565605273397221600, 4.48576209360325789297312885526, 5.49628077301005674781360028304, 7.52091068808960772289628326952, 8.324796826494764068551514324559, 8.786931194290833006559128038298, 10.92315118072156934307382996948, 10.99632573201437702550357952641, 12.48626235146583554714277672207
