| L(s) = 1 | + 5-s − 4·11-s + 2·13-s − 2·17-s + 23-s + 25-s + 6·29-s − 6·37-s − 2·41-s + 12·43-s − 7·49-s − 10·53-s − 4·55-s + 12·59-s − 6·61-s + 2·65-s + 12·67-s + 12·71-s + 2·73-s + 16·79-s − 16·83-s − 2·85-s − 14·89-s + 6·97-s − 10·101-s + 8·103-s + 10·109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 1.11·29-s − 0.986·37-s − 0.312·41-s + 1.82·43-s − 49-s − 1.37·53-s − 0.539·55-s + 1.56·59-s − 0.768·61-s + 0.248·65-s + 1.46·67-s + 1.42·71-s + 0.234·73-s + 1.80·79-s − 1.75·83-s − 0.216·85-s − 1.48·89-s + 0.609·97-s − 0.995·101-s + 0.788·103-s + 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936114224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936114224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.960972345802583748236505828552, −7.01084779076761750159357176047, −6.47911974515012984213798613717, −5.66837052279709760331333880697, −5.09224079361795035060815253989, −4.37260352652791807998889734777, −3.38434121556416518004393820067, −2.63608801407211173598861341226, −1.85232171975255725425548767101, −0.67515771210227546855419498872,
0.67515771210227546855419498872, 1.85232171975255725425548767101, 2.63608801407211173598861341226, 3.38434121556416518004393820067, 4.37260352652791807998889734777, 5.09224079361795035060815253989, 5.66837052279709760331333880697, 6.47911974515012984213798613717, 7.01084779076761750159357176047, 7.960972345802583748236505828552
