| L(s) = 1 | − 4-s + 22·7-s + 10·13-s − 15·16-s − 38·19-s + 14·25-s − 22·28-s − 26·31-s + 34·37-s + 58·43-s + 265·49-s − 10·52-s − 44·61-s + 31·64-s + 196·67-s + 76·73-s + 38·76-s + 94·79-s + 220·91-s + 166·97-s − 14·100-s − 308·103-s − 110·109-s − 330·112-s + 98·121-s + 26·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/4·4-s + 22/7·7-s + 0.769·13-s − 0.937·16-s − 2·19-s + 0.559·25-s − 0.785·28-s − 0.838·31-s + 0.918·37-s + 1.34·43-s + 5.40·49-s − 0.192·52-s − 0.721·61-s + 0.484·64-s + 2.92·67-s + 1.04·73-s + 1/2·76-s + 1.18·79-s + 2.41·91-s + 1.71·97-s − 0.139·100-s − 2.99·103-s − 1.00·109-s − 2.94·112-s + 0.809·121-s + 0.209·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.810666794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810666794\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )( 1 + 8 T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 98 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 622 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2462 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3842 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4322 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6926 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 98 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 9422 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15518 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )^{2} \) |
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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
−12.42842708643973767130047860883, −11.27111905027396325696564942146, −11.16567393233495056098395985919, −10.90512649461616121833257133912, −10.59224937966225158628329408265, −9.583105729012793486951211549718, −9.061797322800659243002030888891, −8.572787252775755938065147406919, −8.116819103211307537777524517370, −8.045726384580780932483356291653, −7.21317449268238264211180967157, −6.65685774015650989618206953334, −5.89470376189093228989638208840, −5.28468111134822449992109182189, −4.68882000533053757128112187773, −4.39643892250480922489295932345, −3.81024637270885270033681341068, −2.28736476258828690753839062426, −1.97366539924764959346726740312, −0.971667461786487855396551709766,
0.971667461786487855396551709766, 1.97366539924764959346726740312, 2.28736476258828690753839062426, 3.81024637270885270033681341068, 4.39643892250480922489295932345, 4.68882000533053757128112187773, 5.28468111134822449992109182189, 5.89470376189093228989638208840, 6.65685774015650989618206953334, 7.21317449268238264211180967157, 8.045726384580780932483356291653, 8.116819103211307537777524517370, 8.572787252775755938065147406919, 9.061797322800659243002030888891, 9.583105729012793486951211549718, 10.59224937966225158628329408265, 10.90512649461616121833257133912, 11.16567393233495056098395985919, 11.27111905027396325696564942146, 12.42842708643973767130047860883
