| L(s) = 1 | − 2·4-s − 3·5-s − 4·7-s + 3·11-s − 13-s + 4·16-s + 17-s − 19-s + 6·20-s − 9·23-s + 4·25-s + 8·28-s − 6·29-s + 2·31-s + 12·35-s − 4·37-s + 3·41-s − 7·43-s − 6·44-s + 6·47-s + 9·49-s + 2·52-s + 6·53-s − 9·55-s − 6·59-s + 8·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s − 1.51·7-s + 0.904·11-s − 0.277·13-s + 16-s + 0.242·17-s − 0.229·19-s + 1.34·20-s − 1.87·23-s + 4/5·25-s + 1.51·28-s − 1.11·29-s + 0.359·31-s + 2.02·35-s − 0.657·37-s + 0.468·41-s − 1.06·43-s − 0.904·44-s + 0.875·47-s + 9/7·49-s + 0.277·52-s + 0.824·53-s − 1.21·55-s − 0.781·59-s + 1.02·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.40008400828974321311560987755, −11.81013235135078842473393257576, −10.23122640158841960226025987177, −9.392750473138958647961697264371, −8.382258919016960888464931958518, −7.25132554126425354900557687617, −5.94155321226588727412442600100, −4.19114262883338708198809215469, −3.52343415212056409865262439727, 0,
3.52343415212056409865262439727, 4.19114262883338708198809215469, 5.94155321226588727412442600100, 7.25132554126425354900557687617, 8.382258919016960888464931958518, 9.392750473138958647961697264371, 10.23122640158841960226025987177, 11.81013235135078842473393257576, 12.40008400828974321311560987755
