| L(s) = 1 | − 2·2-s + 4-s + 12·5-s − 2·7-s − 2·8-s − 3·9-s − 24·10-s − 2·11-s + 4·14-s + 7·16-s − 2·17-s + 6·18-s + 10·19-s + 12·20-s + 4·22-s + 10·23-s + 81·25-s − 2·28-s + 6·29-s − 10·32-s + 4·34-s − 24·35-s − 3·36-s + 8·37-s − 20·38-s − 24·40-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 5.36·5-s − 0.755·7-s − 0.707·8-s − 9-s − 7.58·10-s − 0.603·11-s + 1.06·14-s + 7/4·16-s − 0.485·17-s + 1.41·18-s + 2.29·19-s + 2.68·20-s + 0.852·22-s + 2.08·23-s + 81/5·25-s − 0.377·28-s + 1.11·29-s − 1.76·32-s + 0.685·34-s − 4.05·35-s − 1/2·36-s + 1.31·37-s − 3.24·38-s − 3.79·40-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258065658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258065658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + 3 T^{2} + 3 p T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| good | 7 | \( 1 + 2 T + 2 T^{2} - 18 T^{3} - p^{2} T^{4} + 4 p T^{5} + 316 T^{6} + 4 p^{2} T^{7} - p^{4} T^{8} - 18 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 2 T + 2 T^{2} - 10 T^{3} - 57 T^{4} + 252 T^{5} + 668 T^{6} + 252 p T^{7} - 57 p^{2} T^{8} - 10 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + p T^{2} )^{6} \) |
| 17 | \( 1 + 2 T + 2 T^{2} - 30 T^{3} - p^{2} T^{4} - 4 p T^{5} + 892 T^{6} - 4 p^{2} T^{7} - p^{4} T^{8} - 30 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 10 T + 50 T^{2} - 222 T^{3} + 215 T^{4} + 2836 T^{5} - 14468 T^{6} + 2836 p T^{7} + 215 p^{2} T^{8} - 222 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 10 T + 50 T^{2} - 262 T^{3} + 495 T^{4} + 2100 T^{5} - 11428 T^{6} + 2100 p T^{7} + 495 p^{2} T^{8} - 262 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{3} \) |
| 31 | \( 1 - 126 T^{2} + 7727 T^{4} - 296420 T^{6} + 7727 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 4 T + 47 T^{2} - 168 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{3} \) |
| 43 | \( ( 1 - 2 T + 69 T^{2} - 308 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 10 T + 50 T^{2} + 438 T^{3} - p^{2} T^{4} - 908 p T^{5} - 220388 T^{6} - 908 p^{2} T^{7} - p^{4} T^{8} + 438 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 82 T^{2} + 6231 T^{4} - 379228 T^{6} + 6231 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 6 T + 18 T^{2} + 574 T^{3} - 3417 T^{4} - 20788 T^{5} + 350972 T^{6} - 20788 p T^{7} - 3417 p^{2} T^{8} + 574 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 14 T + 98 T^{2} + 918 T^{3} + 10679 T^{4} + 85828 T^{5} + 576412 T^{6} + 85828 p T^{7} + 10679 p^{2} T^{8} + 918 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 18 T + 173 T^{2} + 1204 T^{3} + 173 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 8 T + 117 T^{2} + 624 T^{3} + 117 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 10 T + 50 T^{2} + 198 T^{3} - 4753 T^{4} + 148 p T^{5} + 28 p^{2} T^{6} + 148 p^{2} T^{7} - 4753 p^{2} T^{8} + 198 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 8 T + 141 T^{2} + 752 T^{3} + 141 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 130 T^{2} + 9575 T^{4} - 446268 T^{6} + 9575 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 10 T + 50 T^{2} + 426 T^{3} - 8833 T^{4} - 142708 T^{5} - 894692 T^{6} - 142708 p T^{7} - 8833 p^{2} T^{8} + 426 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.36555237035231068379734697955, −6.33102255294028323091397249236, −6.19472071589874710036908690982, −6.18085284722408988204481969286, −6.09540787976338711466223165054, −5.83456325091677680283550862036, −5.60726001679410122088829646426, −5.41090159114756043260711454564, −5.09616190009146090910287072466, −5.06137882150267679840471957123, −4.96417898086752453972298993880, −4.95550116810802078471031991063, −4.30828722235258868715092327822, −4.16591727477157283696935163754, −3.34182559848505706546835758761, −3.32178592957981652547342159481, −3.13544126789118907802591041832, −2.79117931033563116338646249625, −2.65113059312843183951537522642, −2.54311772509819876349100080780, −2.42401078885898455519860842618, −1.60239297080772387835749496949, −1.48934801627830924969152753553, −1.37021955175760854068079651177, −0.805116841095489270475009114442,
0.805116841095489270475009114442, 1.37021955175760854068079651177, 1.48934801627830924969152753553, 1.60239297080772387835749496949, 2.42401078885898455519860842618, 2.54311772509819876349100080780, 2.65113059312843183951537522642, 2.79117931033563116338646249625, 3.13544126789118907802591041832, 3.32178592957981652547342159481, 3.34182559848505706546835758761, 4.16591727477157283696935163754, 4.30828722235258868715092327822, 4.95550116810802078471031991063, 4.96417898086752453972298993880, 5.06137882150267679840471957123, 5.09616190009146090910287072466, 5.41090159114756043260711454564, 5.60726001679410122088829646426, 5.83456325091677680283550862036, 6.09540787976338711466223165054, 6.18085284722408988204481969286, 6.19472071589874710036908690982, 6.33102255294028323091397249236, 6.36555237035231068379734697955
Plot not available for L-functions of degree greater than 10.
