| L(s) = 1 | + 2·2-s + 4·4-s − 8·5-s + 8·8-s − 16·10-s − 10·13-s + 16·16-s + 16·17-s − 32·20-s + 39·25-s − 20·26-s + 40·29-s + 32·32-s + 32·34-s − 70·37-s − 64·40-s − 80·41-s + 49·49-s + 78·50-s − 40·52-s − 56·53-s + 80·58-s − 22·61-s + 64·64-s + 80·65-s + 64·68-s + 110·73-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 8/5·5-s + 8-s − 8/5·10-s − 0.769·13-s + 16-s + 0.941·17-s − 8/5·20-s + 1.55·25-s − 0.769·26-s + 1.37·29-s + 32-s + 0.941·34-s − 1.89·37-s − 8/5·40-s − 1.95·41-s + 49-s + 1.55·50-s − 0.769·52-s − 1.05·53-s + 1.37·58-s − 0.360·61-s + 64-s + 1.23·65-s + 0.941·68-s + 1.50·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.465904636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465904636\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 10 T + p^{2} T^{2} \) |
| 17 | \( 1 - 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 70 T + p^{2} T^{2} \) |
| 41 | \( 1 + 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 160 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
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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
−15.87270527187283585157096256752, −15.11660626150141738993593107084, −13.99957549541272776025518168753, −12.34562003886530270901028120653, −11.84733715210479263681904402850, −10.44912621711228457712744653491, −8.114225158872954116270587294830, −6.97680605167142446406053986467, −4.91527951657648063383825038644, −3.42919429089720191179666185785,
3.42919429089720191179666185785, 4.91527951657648063383825038644, 6.97680605167142446406053986467, 8.114225158872954116270587294830, 10.44912621711228457712744653491, 11.84733715210479263681904402850, 12.34562003886530270901028120653, 13.99957549541272776025518168753, 15.11660626150141738993593107084, 15.87270527187283585157096256752
