| L(s) = 1 | − 16·4-s − 23·7-s − 146·13-s − 92·19-s − 625·25-s + 368·28-s + 1.75e3·31-s + 5.18e3·37-s − 3.19e3·43-s + 2.40e3·49-s + 2.33e3·52-s − 7.19e3·61-s + 4.09e3·64-s − 2.90e3·67-s − 2.49e3·73-s + 1.47e3·76-s − 7.68e3·79-s + 3.35e3·91-s + 1.88e4·97-s + 1.00e4·100-s + 1.98e4·103-s − 3.74e4·109-s − 1.46e4·121-s − 2.80e4·124-s + 127-s + 131-s + 2.11e3·133-s + ⋯ |
| L(s) = 1 | − 4-s − 0.469·7-s − 0.863·13-s − 0.254·19-s − 25-s + 0.469·28-s + 1.82·31-s + 3.78·37-s − 1.72·43-s + 49-s + 0.863·52-s − 1.93·61-s + 64-s − 0.646·67-s − 0.468·73-s + 0.254·76-s − 1.23·79-s + 0.405·91-s + 1.99·97-s + 100-s + 1.87·103-s − 3.15·109-s − 121-s − 1.82·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 0.119·133-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7496276412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7496276412\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - 71 T + p^{4} T^{2} )( 1 + 94 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 191 T + p^{4} T^{2} )( 1 + 337 T + p^{4} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 46 T + p^{4} T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 1559 T + p^{4} T^{2} )( 1 - 194 T + p^{4} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2591 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 23 T + p^{4} T^{2} )( 1 + 3214 T + p^{4} T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4679 T + p^{4} T^{2} )( 1 + 12361 T + p^{4} T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 9743 T + p^{4} T^{2} )( 1 - 9071 T + p^{4} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.60397748532734674597024114716, −11.47567724407765623604245094692, −10.54290144237283085284603294092, −10.10451562154784742473323641233, −9.762093323766747811399242705516, −9.307354713378154205468378828913, −8.919013655180644965803006102464, −8.230923752294517457024204820278, −7.79826327331685303568977491015, −7.38677374662566237286461706916, −6.41219359930857586924263765151, −6.26234285873435095333630821921, −5.50495017690024806443037388710, −4.67180743543497273578921704984, −4.49040353374568817737976973445, −3.83718960186796514941340592939, −2.87291860333793393985612770730, −2.41692641389212693770415351367, −1.21499610502172753493221195579, −0.31866828326953643968611324011,
0.31866828326953643968611324011, 1.21499610502172753493221195579, 2.41692641389212693770415351367, 2.87291860333793393985612770730, 3.83718960186796514941340592939, 4.49040353374568817737976973445, 4.67180743543497273578921704984, 5.50495017690024806443037388710, 6.26234285873435095333630821921, 6.41219359930857586924263765151, 7.38677374662566237286461706916, 7.79826327331685303568977491015, 8.230923752294517457024204820278, 8.919013655180644965803006102464, 9.307354713378154205468378828913, 9.762093323766747811399242705516, 10.10451562154784742473323641233, 10.54290144237283085284603294092, 11.47567724407765623604245094692, 11.60397748532734674597024114716
