| L(s) = 1 | − 2·5-s − 7-s + 6·11-s + 6·13-s − 2·17-s + 4·19-s − 2·23-s − 25-s − 8·29-s − 4·31-s + 2·35-s + 6·37-s + 10·41-s − 4·43-s + 4·47-s + 49-s + 4·53-s − 12·55-s − 12·59-s + 2·61-s − 12·65-s + 12·67-s − 6·71-s − 2·73-s − 6·77-s + 8·79-s + 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.80·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.417·23-s − 1/5·25-s − 1.48·29-s − 0.718·31-s + 0.338·35-s + 0.986·37-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.549·53-s − 1.61·55-s − 1.56·59-s + 0.256·61-s − 1.48·65-s + 1.46·67-s − 0.712·71-s − 0.234·73-s − 0.683·77-s + 0.900·79-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773018634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773018634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.481230422127386542430938775743, −7.67675156505404098748222537011, −7.02225335608143008119808893311, −6.18904954467597583465872191621, −5.72096147163059808933542492052, −4.29551882482260666462775014234, −3.86692381983966648059599058083, −3.29743767977860739452763883672, −1.80228330876561683103867681836, −0.798602691922161062936643959711,
0.798602691922161062936643959711, 1.80228330876561683103867681836, 3.29743767977860739452763883672, 3.86692381983966648059599058083, 4.29551882482260666462775014234, 5.72096147163059808933542492052, 6.18904954467597583465872191621, 7.02225335608143008119808893311, 7.67675156505404098748222537011, 8.481230422127386542430938775743
