| L(s) = 1 | + 9·3-s + 16·4-s + 81·9-s + 144·12-s + 191·13-s + 256·16-s − 601·19-s + 625·25-s + 729·27-s − 1.75e3·31-s + 1.29e3·36-s + 2.59e3·37-s + 1.71e3·39-s + 23·43-s + 2.30e3·48-s + 3.05e3·52-s − 5.40e3·57-s − 1.96e3·61-s + 4.09e3·64-s − 8.80e3·67-s − 1.24e3·73-s + 5.62e3·75-s − 9.61e3·76-s − 1.23e4·79-s + 6.56e3·81-s − 1.57e4·93-s − 1.88e4·97-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 9-s + 12-s + 1.13·13-s + 16-s − 1.66·19-s + 25-s + 27-s − 1.82·31-s + 36-s + 1.89·37-s + 1.13·39-s + 0.0124·43-s + 48-s + 1.13·52-s − 1.66·57-s − 0.528·61-s + 64-s − 1.96·67-s − 0.234·73-s + 75-s − 1.66·76-s − 1.98·79-s + 81-s − 1.82·93-s − 1.99·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.331662218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.331662218\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 191 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 601 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 1753 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2591 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 23 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 1966 T + p^{4} T^{2} \) |
| 67 | \( 1 + 8809 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 1249 T + p^{4} T^{2} \) |
| 79 | \( 1 + 12361 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 + 18814 T + p^{4} T^{2} \) |
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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
−12.58104607114060352997339651146, −11.15494702198054681404899311274, −10.46926812420951228722342111984, −9.095388278492241886155572876093, −8.179754775476964735111059709801, −7.09312330997744730265845564815, −6.05663936391500956845734084108, −4.11439103484865717582566679473, −2.83301770432692380350337036688, −1.56411795166496279058804682706,
1.56411795166496279058804682706, 2.83301770432692380350337036688, 4.11439103484865717582566679473, 6.05663936391500956845734084108, 7.09312330997744730265845564815, 8.179754775476964735111059709801, 9.095388278492241886155572876093, 10.46926812420951228722342111984, 11.15494702198054681404899311274, 12.58104607114060352997339651146
