| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 14-s − 16-s − 8·17-s − 5·23-s + 3·25-s + 28-s + 4·31-s − 5·32-s + 8·34-s − 10·41-s + 5·46-s + 3·47-s − 11·49-s − 3·50-s − 3·56-s − 4·62-s + 7·64-s + 8·68-s − 9·71-s + 6·73-s − 13·79-s + 10·82-s − 89-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.267·14-s − 1/4·16-s − 1.94·17-s − 1.04·23-s + 3/5·25-s + 0.188·28-s + 0.718·31-s − 0.883·32-s + 1.37·34-s − 1.56·41-s + 0.737·46-s + 0.437·47-s − 1.57·49-s − 0.424·50-s − 0.400·56-s − 0.508·62-s + 7/8·64-s + 0.970·68-s − 1.06·71-s + 0.702·73-s − 1.46·79-s + 1.10·82-s − 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.845500915786901372532501424148, −9.392924781146833517009841345505, −8.783092352182644861930438254241, −8.533343644276970174588719120067, −7.985135492659449456228284043791, −7.36163256647035612276638205075, −6.65936899910384011197839051632, −6.40820004047362281446656683773, −5.49343550173419062808987775443, −4.77241224410410548152924719575, −4.34298347992956524752722635530, −3.60825846460810150961976756718, −2.60669338612344229328052534848, −1.62458108659373753119003009659, 0,
1.62458108659373753119003009659, 2.60669338612344229328052534848, 3.60825846460810150961976756718, 4.34298347992956524752722635530, 4.77241224410410548152924719575, 5.49343550173419062808987775443, 6.40820004047362281446656683773, 6.65936899910384011197839051632, 7.36163256647035612276638205075, 7.985135492659449456228284043791, 8.533343644276970174588719120067, 8.783092352182644861930438254241, 9.392924781146833517009841345505, 9.845500915786901372532501424148
