| L(s) = 1 | − 16·4-s + 94·7-s − 191·13-s + 1.29e3·19-s − 625·25-s − 1.50e3·28-s − 1.55e3·31-s − 4.12e3·37-s − 23·43-s + 2.40e3·49-s + 3.05e3·52-s − 7.19e3·61-s + 4.09e3·64-s − 2.90e3·67-s − 2.49e3·73-s − 2.07e4·76-s + 1.23e4·79-s − 1.79e4·91-s − 9.74e3·97-s + 1.00e4·100-s + 1.98e4·103-s − 6.62e3·109-s − 1.46e4·121-s + 2.49e4·124-s + 127-s + 131-s + 1.21e5·133-s + ⋯ |
| L(s) = 1 | − 4-s + 1.91·7-s − 1.13·13-s + 3.58·19-s − 25-s − 1.91·28-s − 1.62·31-s − 3.01·37-s − 0.0124·43-s + 49-s + 1.13·52-s − 1.93·61-s + 64-s − 0.646·67-s − 0.468·73-s − 3.58·76-s + 1.98·79-s − 2.16·91-s − 1.03·97-s + 100-s + 1.87·103-s − 0.557·109-s − 121-s + 1.62·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 6.87·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.782845358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782845358\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 71 T + p^{4} T^{2} )( 1 - 23 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 146 T + p^{4} T^{2} )( 1 + 337 T + p^{4} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 647 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 194 T + p^{4} T^{2} )( 1 + 1753 T + p^{4} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2062 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3191 T + p^{4} T^{2} )( 1 + 3214 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 1966 T + p^{4} T^{2} )( 1 + 5233 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5906 T + p^{4} T^{2} )( 1 + 8809 T + p^{4} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1249 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 7682 T + p^{4} T^{2} )( 1 - 4679 T + p^{4} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9071 T + p^{4} T^{2} )( 1 + 18814 T + p^{4} T^{2} ) \) |
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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
−12.09892049095389389262054083715, −11.27031053686909232114357432012, −10.81150459510737482128690534597, −10.18261480244659231268064752490, −9.614798788478988659345006792755, −9.292918297100647760308271362308, −8.899119332520387901427108030087, −8.123069594090939101841089656356, −7.83501846035774943238839697607, −7.25153255993842633663552361537, −7.07262121414069284740866195571, −5.60842033065437838288013274529, −5.47491169138528147453056394111, −4.81594117782555890264775959962, −4.70870916954877741044367179329, −3.63442364160275491386454673031, −3.17831865652687001881434225854, −1.91356835086187746802474357698, −1.50770018003628127318744858611, −0.46255163489629475090197984647,
0.46255163489629475090197984647, 1.50770018003628127318744858611, 1.91356835086187746802474357698, 3.17831865652687001881434225854, 3.63442364160275491386454673031, 4.70870916954877741044367179329, 4.81594117782555890264775959962, 5.47491169138528147453056394111, 5.60842033065437838288013274529, 7.07262121414069284740866195571, 7.25153255993842633663552361537, 7.83501846035774943238839697607, 8.123069594090939101841089656356, 8.899119332520387901427108030087, 9.292918297100647760308271362308, 9.614798788478988659345006792755, 10.18261480244659231268064752490, 10.81150459510737482128690534597, 11.27031053686909232114357432012, 12.09892049095389389262054083715
