| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s − 11-s + 6·13-s − 2·15-s − 4·17-s − 2·19-s − 2·21-s − 8·23-s − 25-s − 27-s + 33-s + 4·35-s − 6·37-s − 6·39-s + 10·43-s + 2·45-s − 3·49-s + 4·51-s + 14·53-s − 2·55-s + 2·57-s − 12·59-s − 14·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 0.970·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.174·33-s + 0.676·35-s − 0.986·37-s − 0.960·39-s + 1.52·43-s + 0.298·45-s − 3/7·49-s + 0.560·51-s + 1.92·53-s − 0.269·55-s + 0.264·57-s − 1.56·59-s − 1.79·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.094444217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094444217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.45826335615790311268939932306, −12.22139625462217175200876644011, −11.07464839362546140922098622524, −10.44502381868694291876952676407, −9.106296123632974974434793663066, −8.017674089261979349140094913788, −6.41893627610765861044252902088, −5.62860857478680063008223727075, −4.18020563951979381603420573914, −1.86785039249087723872477729694,
1.86785039249087723872477729694, 4.18020563951979381603420573914, 5.62860857478680063008223727075, 6.41893627610765861044252902088, 8.017674089261979349140094913788, 9.106296123632974974434793663066, 10.44502381868694291876952676407, 11.07464839362546140922098622524, 12.22139625462217175200876644011, 13.45826335615790311268939932306
