| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s − 4·10-s + 2·11-s + 13-s − 16-s − 6·17-s + 4·19-s + 4·20-s + 2·22-s + 6·23-s + 11·25-s + 26-s + 2·29-s + 3·31-s + 5·32-s − 6·34-s + 3·37-s + 4·38-s + 12·40-s − 2·41-s − 43-s − 2·44-s + 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 1.26·10-s + 0.603·11-s + 0.277·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.894·20-s + 0.426·22-s + 1.25·23-s + 11/5·25-s + 0.196·26-s + 0.371·29-s + 0.538·31-s + 0.883·32-s − 1.02·34-s + 0.493·37-s + 0.648·38-s + 1.89·40-s − 0.312·41-s − 0.152·43-s − 0.301·44-s + 0.884·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218539476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218539476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−9.356883354769464972357969524740, −8.785111346731224547228439173752, −8.067909489204395286784670333572, −7.11526166228041896275075317849, −6.37671245558863001404255472344, −5.05817222407614461034531924213, −4.43082498076330884740103768112, −3.72686733644268071328326458986, −2.92845750765415073814212919525, −0.72695453395181431252401129496,
0.72695453395181431252401129496, 2.92845750765415073814212919525, 3.72686733644268071328326458986, 4.43082498076330884740103768112, 5.05817222407614461034531924213, 6.37671245558863001404255472344, 7.11526166228041896275075317849, 8.067909489204395286784670333572, 8.785111346731224547228439173752, 9.356883354769464972357969524740
