| L(s) = 1 | + 2-s − 6·3-s + 4-s − 6·6-s + 8-s + 21·9-s − 6·12-s + 8·13-s + 16-s + 21·18-s − 6·24-s + 25-s + 8·26-s − 56·27-s − 16·31-s + 5·32-s + 21·36-s + 16·37-s − 48·39-s − 4·41-s − 6·48-s + 18·49-s + 50-s + 8·52-s − 24·53-s − 56·54-s − 16·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3.46·3-s + 1/2·4-s − 2.44·6-s + 0.353·8-s + 7·9-s − 1.73·12-s + 2.21·13-s + 1/4·16-s + 4.94·18-s − 1.22·24-s + 1/5·25-s + 1.56·26-s − 10.7·27-s − 2.87·31-s + 0.883·32-s + 7/2·36-s + 2.63·37-s − 7.68·39-s − 0.624·41-s − 0.866·48-s + 18/7·49-s + 0.141·50-s + 1.10·52-s − 3.29·53-s − 7.62·54-s − 2.03·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6196423910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6196423910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T - p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | \( ( 1 + T )^{6} \) |
| 5 | \( 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 4 T + 23 T^{2} - 48 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} + 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 8 T + 185 T^{2} - 880 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 10 T + 103 T^{2} + 396 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41723458285377844873052520256, −7.23918524644620620596748114809, −7.05349848888652327567376920367, −6.86981232137470216174228105715, −6.63721449314990582705857205057, −6.22691155196623033128677063676, −6.15148969379611355540884927692, −6.14296126560268722196977971321, −6.07284318511331767459595709353, −5.69149156033938695983616953089, −5.42270100349333544655125648144, −5.33880860225500696184261446878, −5.06683614032494353272679440030, −4.87299565805114013270565354649, −4.41879866932439474026769339950, −4.39508971616457155186481528911, −4.22098229968495723923824090517, −3.74240884645171283901061950817, −3.48817147279504785598946044861, −3.46030293882061327036993626990, −2.87615994388096863389644470442, −2.04128664062871564010657097566, −2.03227801394467690408670626568, −1.24970307222334753202882376774, −0.884039741002023211256200436812,
0.884039741002023211256200436812, 1.24970307222334753202882376774, 2.03227801394467690408670626568, 2.04128664062871564010657097566, 2.87615994388096863389644470442, 3.46030293882061327036993626990, 3.48817147279504785598946044861, 3.74240884645171283901061950817, 4.22098229968495723923824090517, 4.39508971616457155186481528911, 4.41879866932439474026769339950, 4.87299565805114013270565354649, 5.06683614032494353272679440030, 5.33880860225500696184261446878, 5.42270100349333544655125648144, 5.69149156033938695983616953089, 6.07284318511331767459595709353, 6.14296126560268722196977971321, 6.15148969379611355540884927692, 6.22691155196623033128677063676, 6.63721449314990582705857205057, 6.86981232137470216174228105715, 7.05349848888652327567376920367, 7.23918524644620620596748114809, 7.41723458285377844873052520256
Plot not available for L-functions of degree greater than 10.
