| L(s) = 1 | − 2·5-s + 11-s + 2·13-s − 6·17-s + 4·23-s − 25-s − 2·29-s − 10·37-s − 6·41-s + 8·43-s − 4·47-s − 7·49-s + 6·53-s − 2·55-s − 12·59-s + 2·61-s − 4·65-s − 4·67-s + 12·71-s − 14·73-s − 16·79-s − 12·83-s + 12·85-s − 10·89-s − 14·97-s + 6·101-s − 8·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s − 49-s + 0.824·53-s − 0.269·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 1.42·71-s − 1.63·73-s − 1.80·79-s − 1.31·83-s + 1.30·85-s − 1.05·89-s − 1.42·97-s + 0.597·101-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.810168097313206799366926878295, −8.404961080795805762995868210790, −7.32016744932745984183700903255, −6.77466584995588965139403496717, −5.77243817256599415521793879046, −4.67845634196735523405640953456, −3.95621199715159929211673824139, −3.03252218576564571484914000812, −1.64726707215079580467207130554, 0,
1.64726707215079580467207130554, 3.03252218576564571484914000812, 3.95621199715159929211673824139, 4.67845634196735523405640953456, 5.77243817256599415521793879046, 6.77466584995588965139403496717, 7.32016744932745984183700903255, 8.404961080795805762995868210790, 8.810168097313206799366926878295
