| L(s) = 1 | − 9·2-s + 38·4-s + 45·5-s − 50·7-s − 99·8-s − 405·10-s − 36·11-s − 254·13-s + 450·14-s + 297·16-s + 502·19-s + 1.71e3·20-s + 324·22-s + 585·23-s + 725·25-s + 2.28e3·26-s − 1.90e3·28-s + 1.88e3·29-s − 380·31-s − 1.58e3·32-s − 2.25e3·35-s − 2.64e3·37-s − 4.51e3·38-s − 4.45e3·40-s + 5.36e3·41-s + 3.58e3·43-s − 1.36e3·44-s + ⋯ |
| L(s) = 1 | − 9/4·2-s + 19/8·4-s + 9/5·5-s − 1.02·7-s − 1.54·8-s − 4.04·10-s − 0.297·11-s − 1.50·13-s + 2.29·14-s + 1.16·16-s + 1.39·19-s + 4.27·20-s + 0.669·22-s + 1.10·23-s + 1.15·25-s + 3.38·26-s − 2.42·28-s + 2.23·29-s − 0.395·31-s − 1.54·32-s − 1.83·35-s − 1.93·37-s − 3.12·38-s − 2.78·40-s + 3.19·41-s + 1.93·43-s − 0.706·44-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8858176120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8858176120\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + 9 T + 43 T^{2} + 9 p^{4} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 p T + 52 p^{2} T^{2} - 9 p^{5} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 50 T + 99 T^{2} + 50 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 36 T + 15073 T^{2} + 36 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 254 T + 35955 T^{2} + 254 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 49430 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 251 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 585 T + 393916 T^{2} - 585 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1881 T + 1886668 T^{2} - 1881 p^{4} T^{3} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 380 T - 779121 T^{2} + 380 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 1324 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5364 T + 12416593 T^{2} - 5364 p^{4} T^{3} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3586 T + 9440595 T^{2} - 3586 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2421 T + 6833428 T^{2} - 2421 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9570985 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3114 T + 15349693 T^{2} + 3114 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1826 T - 10511565 T^{2} + 1826 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5225 T + 7149504 T^{2} + 5225 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 37533625 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3031 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 4360 T - 19940481 T^{2} - 4360 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11700 T + 93088321 T^{2} + 11700 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 16401782 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 319 T - 88427520 T^{2} - 319 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−11.44241559138872609119673272761, −10.73932817981786396518100268794, −10.23716990524739618907458193191, −10.18831987538648108861407852822, −9.693742489765305891429182073703, −9.235154418345490687687454077394, −9.040500006858140472670510515123, −8.707895237534129095778424629615, −7.57201214809342110178711634561, −7.49165815931875984849945546551, −7.02678884804736441023691836919, −6.09219529057130851156642494931, −5.87775647390409440213741853092, −5.22250488063369782037058402242, −4.44370322383923709426988749433, −2.99968844633303426146232767889, −2.75269242673847530559704794722, −1.94867965691724443493220103711, −1.01040573679681058792750597496, −0.56535294344395599633787371388,
0.56535294344395599633787371388, 1.01040573679681058792750597496, 1.94867965691724443493220103711, 2.75269242673847530559704794722, 2.99968844633303426146232767889, 4.44370322383923709426988749433, 5.22250488063369782037058402242, 5.87775647390409440213741853092, 6.09219529057130851156642494931, 7.02678884804736441023691836919, 7.49165815931875984849945546551, 7.57201214809342110178711634561, 8.707895237534129095778424629615, 9.040500006858140472670510515123, 9.235154418345490687687454077394, 9.693742489765305891429182073703, 10.18831987538648108861407852822, 10.23716990524739618907458193191, 10.73932817981786396518100268794, 11.44241559138872609119673272761
