| L(s) = 1 | + 64·4-s + 683·7-s + 3.52e3·13-s + 4.09e3·16-s − 2.26e3·19-s + 1.56e4·25-s + 4.37e4·28-s − 5.92e4·31-s + 3.02e3·37-s − 1.53e5·43-s + 3.48e5·49-s + 2.25e5·52-s − 4.20e5·61-s + 2.62e5·64-s + 1.72e5·67-s + 6.38e5·73-s − 1.45e5·76-s − 7.33e5·79-s + 2.40e6·91-s + 1.60e6·97-s + 1.00e6·100-s + 1.12e6·103-s − 1.34e5·109-s + 2.79e6·112-s + ⋯ |
| L(s) = 1 | + 4-s + 1.99·7-s + 1.60·13-s + 16-s − 0.330·19-s + 25-s + 1.99·28-s − 1.98·31-s + 0.0596·37-s − 1.93·43-s + 2.96·49-s + 1.60·52-s − 1.85·61-s + 64-s + 0.574·67-s + 1.64·73-s − 0.330·76-s − 1.48·79-s + 3.19·91-s + 1.76·97-s + 100-s + 1.03·103-s − 0.103·109-s + 1.99·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.914945511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.914945511\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 - 683 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 3527 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 + 2269 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 + 59221 T + p^{6} T^{2} \) |
| 37 | \( 1 - 3023 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 + 153973 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 420838 T + p^{6} T^{2} \) |
| 67 | \( 1 - 172874 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 638066 T + p^{6} T^{2} \) |
| 79 | \( 1 + 733069 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 - 1608263 T + p^{6} T^{2} \) |
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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
−11.07966753464351819712385130111, −10.56398450277929876092233643508, −8.809509925382378112625204423949, −8.102016871504924403262673058347, −7.14484284260173000057381453603, −5.96489751490782875554400634319, −4.91925597754285058400977041564, −3.54756835987430544575636903139, −1.96149843421224193380747139469, −1.24396368371873547203781654379,
1.24396368371873547203781654379, 1.96149843421224193380747139469, 3.54756835987430544575636903139, 4.91925597754285058400977041564, 5.96489751490782875554400634319, 7.14484284260173000057381453603, 8.102016871504924403262673058347, 8.809509925382378112625204423949, 10.56398450277929876092233643508, 11.07966753464351819712385130111
