| L(s) = 1 | − 128·4-s − 729·9-s + 1.22e4·16-s + 2.11e4·19-s + 7.05e4·31-s + 9.33e4·36-s + 1.53e5·49-s − 8.41e5·61-s − 1.04e6·64-s − 2.70e6·76-s + 4.09e5·79-s + 5.31e5·81-s + 4.34e6·109-s + 3.54e6·121-s − 9.03e6·124-s + 127-s + 131-s + 137-s + 139-s − 8.95e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.39e6·169-s − 1.54e7·171-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 3.08·19-s + 2.36·31-s + 2·36-s + 1.30·49-s − 3.70·61-s − 4·64-s − 6.17·76-s + 0.830·79-s + 81-s + 3.35·109-s + 2·121-s − 4.73·124-s − 3·144-s + 1.94·169-s − 3.08·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.334127184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334127184\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 153502 T^{2} + p^{12} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 9397582 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 10582 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 35282 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2826257618 T^{2} + p^{12} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 235885102 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 420838 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 151031344462 T^{2} + p^{12} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 104459767778 T^{2} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 204622 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1662757858942 T^{2} + p^{12} T^{4} \) |
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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
−13.61587216363107662399393547845, −13.45768987889500577519856042939, −12.37682138915530620255834464222, −12.11307510215001099397436042564, −11.58208819437770289268308782701, −10.71927500835144163900893791338, −9.857817741933418514232921276411, −9.773930297832765476522475427445, −8.918601559971055287399416618298, −8.724339128276756680587988049490, −7.74370167361227050436387196653, −7.61802200503554585405045218592, −6.18036236124703964308565672939, −5.61024738007369046445097591718, −4.98282050826968890496107907231, −4.47441192740753461411610003764, −3.38256365635904452590860672978, −2.99090689549102054241367668730, −1.13585494306597733807108741887, −0.56162062531571553231000336270,
0.56162062531571553231000336270, 1.13585494306597733807108741887, 2.99090689549102054241367668730, 3.38256365635904452590860672978, 4.47441192740753461411610003764, 4.98282050826968890496107907231, 5.61024738007369046445097591718, 6.18036236124703964308565672939, 7.61802200503554585405045218592, 7.74370167361227050436387196653, 8.724339128276756680587988049490, 8.918601559971055287399416618298, 9.773930297832765476522475427445, 9.857817741933418514232921276411, 10.71927500835144163900893791338, 11.58208819437770289268308782701, 12.11307510215001099397436042564, 12.37682138915530620255834464222, 13.45768987889500577519856042939, 13.61587216363107662399393547845
