| L(s) = 1 | + 8·4-s + 37·7-s − 89·13-s + 214·19-s + 125·25-s + 296·28-s + 19·31-s + 646·37-s − 71·43-s + 343·49-s − 712·52-s − 182·61-s − 512·64-s + 880·67-s + 2.38e3·73-s + 1.71e3·76-s − 503·79-s − 3.29e3·91-s − 1.85e3·97-s + 1.00e3·100-s − 1.82e3·103-s − 3.13e3·109-s + 1.33e3·121-s + 152·124-s + 127-s + 131-s + 7.91e3·133-s + ⋯ |
| L(s) = 1 | + 4-s + 1.99·7-s − 1.89·13-s + 2.58·19-s + 25-s + 1.99·28-s + 0.110·31-s + 2.87·37-s − 0.251·43-s + 49-s − 1.89·52-s − 0.382·61-s − 64-s + 1.60·67-s + 3.81·73-s + 2.58·76-s − 0.716·79-s − 3.79·91-s − 1.93·97-s + 100-s − 1.74·103-s − 2.75·109-s + 121-s + 0.110·124-s + 0.000698·127-s + 0.000666·131-s + 5.16·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.442497439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.442497439\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 - 17 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 107 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 449 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 523 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \) |
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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
−11.63730525729914053747297894048, −11.47666138466274282454232662439, −11.14139092908553642539513740819, −10.69604002581313491880009790646, −9.876822365786481505342294260471, −9.564918228055938816526468526651, −9.168759272424020900351667585568, −8.108777342445321669279755497386, −7.78526420847137057317431687637, −7.72290391340436919658948115478, −6.87289777273766603673535823315, −6.62693695581247097141506441347, −5.49177493412488091924479351010, −5.15811000801069046026334988686, −4.80164124670713797082005923919, −4.02646502549589841214283453688, −2.78794363479442382405206389291, −2.57111952956869020330969349219, −1.61399786090002047921570212467, −0.903307379252712555608757231174,
0.903307379252712555608757231174, 1.61399786090002047921570212467, 2.57111952956869020330969349219, 2.78794363479442382405206389291, 4.02646502549589841214283453688, 4.80164124670713797082005923919, 5.15811000801069046026334988686, 5.49177493412488091924479351010, 6.62693695581247097141506441347, 6.87289777273766603673535823315, 7.72290391340436919658948115478, 7.78526420847137057317431687637, 8.108777342445321669279755497386, 9.168759272424020900351667585568, 9.564918228055938816526468526651, 9.876822365786481505342294260471, 10.69604002581313491880009790646, 11.14139092908553642539513740819, 11.47666138466274282454232662439, 11.63730525729914053747297894048
