| L(s) = 1 | + 3·2-s − 2·3-s + 4-s − 6·6-s − 7·8-s − 29·9-s + 30·11-s − 2·12-s − 19·16-s + 2·17-s − 87·18-s − 2·19-s + 90·22-s + 14·24-s + 46·27-s − 35·32-s − 60·33-s + 6·34-s − 29·36-s − 6·38-s − 70·41-s − 76·43-s + 30·44-s + 38·48-s + 78·49-s − 4·51-s + 138·54-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 2/3·3-s + 1/4·4-s − 6-s − 7/8·8-s − 3.22·9-s + 2.72·11-s − 1/6·12-s − 1.18·16-s + 2/17·17-s − 4.83·18-s − 0.105·19-s + 4.09·22-s + 7/12·24-s + 1.70·27-s − 1.09·32-s − 1.81·33-s + 3/17·34-s − 0.805·36-s − 0.157·38-s − 1.70·41-s − 1.76·43-s + 0.681·44-s + 0.791·48-s + 1.59·49-s − 0.0784·51-s + 23/9·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.063147270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063147270\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3 T + p^{3} T^{2} - 7 p T^{3} + p^{5} T^{4} - 3 p^{4} T^{5} + p^{6} T^{6} \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 + T + 16 T^{2} + 23 T^{3} + 16 p^{2} T^{4} + p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 7 | \( 1 - 78 T^{2} + 6879 T^{4} - 314724 T^{6} + 6879 p^{4} T^{8} - 78 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( ( 1 - 15 T + 320 T^{2} - 3577 T^{3} + 320 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 398 T^{2} + 116079 T^{4} - 22631204 T^{6} + 116079 p^{4} T^{8} - 398 p^{8} T^{10} + p^{12} T^{12} \) |
| 17 | \( ( 1 - T + 546 T^{2} - 1633 T^{3} + 546 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( ( 1 + T + 32 p T^{2} + 1095 T^{3} + 32 p^{3} T^{4} + p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 38 T^{2} + 249839 T^{4} - 186877524 T^{6} + 249839 p^{4} T^{8} - 38 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 2126 T^{2} + 3163695 T^{4} - 3151366820 T^{6} + 3163695 p^{4} T^{8} - 2126 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( 1 - 166 T^{2} + 422415 T^{4} - 78680020 T^{6} + 422415 p^{4} T^{8} - 166 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 4878 T^{2} + 11381519 T^{4} - 18009516964 T^{6} + 11381519 p^{4} T^{8} - 4878 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( ( 1 + 35 T + 2310 T^{2} + 91223 T^{3} + 2310 p^{2} T^{4} + 35 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( ( 1 + 38 T + 3643 T^{2} + 120524 T^{3} + 3643 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 8078 T^{2} + 32098559 T^{4} - 84067879524 T^{6} + 32098559 p^{4} T^{8} - 8078 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 7318 T^{2} + 23442399 T^{4} - 56470960564 T^{6} + 23442399 p^{4} T^{8} - 7318 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( ( 1 + 22 T + 9339 T^{2} + 159916 T^{3} + 9339 p^{2} T^{4} + 22 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 4526 T^{2} + 43251375 T^{4} - 123231264740 T^{6} + 43251375 p^{4} T^{8} - 4526 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( ( 1 + 9 T + 4856 T^{2} + 62647 T^{3} + 4856 p^{2} T^{4} + 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 10286 T^{2} + 75091935 T^{4} - 75287780 p^{2} T^{6} + 75091935 p^{4} T^{8} - 10286 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 9 T + 5666 T^{2} - 252857 T^{3} + 5666 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 29166 T^{2} + 396253055 T^{4} - 3149992522660 T^{6} + 396253055 p^{4} T^{8} - 29166 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( ( 1 - 199 T + 31136 T^{2} - 2872977 T^{3} + 31136 p^{2} T^{4} - 199 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( ( 1 - 49 T + 21778 T^{2} - 678865 T^{3} + 21778 p^{2} T^{4} - 49 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 - 38 T + 16143 T^{2} - 596564 T^{3} + 16143 p^{2} T^{4} - 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.60335267904141672083898668620, −6.51195174267065284423523367605, −6.25258896819233876180912111500, −5.91010760491596608798696119796, −5.68616447769452323896771647593, −5.66507428563098539036567829540, −5.43287102784741899203976397198, −5.20677552259851695687114554785, −5.01449904480389583696518798952, −4.99126451863599013751430887229, −4.60870731465447296504957921381, −4.34403584900831544881147215393, −4.02744301425780164226418781575, −4.02185577418639376999487789139, −3.65031591695696380148567203233, −3.39095173538828095767986473324, −3.34283721241523518972170618567, −3.19046853114557497453780878649, −2.61879695102050989550981361851, −2.39872053063954967274527525731, −2.22969274396517389683730552041, −1.50043499492623492216483488093, −1.49516759010553905353650719610, −0.59683371595612176262245947675, −0.32981922975272285463104529760,
0.32981922975272285463104529760, 0.59683371595612176262245947675, 1.49516759010553905353650719610, 1.50043499492623492216483488093, 2.22969274396517389683730552041, 2.39872053063954967274527525731, 2.61879695102050989550981361851, 3.19046853114557497453780878649, 3.34283721241523518972170618567, 3.39095173538828095767986473324, 3.65031591695696380148567203233, 4.02185577418639376999487789139, 4.02744301425780164226418781575, 4.34403584900831544881147215393, 4.60870731465447296504957921381, 4.99126451863599013751430887229, 5.01449904480389583696518798952, 5.20677552259851695687114554785, 5.43287102784741899203976397198, 5.66507428563098539036567829540, 5.68616447769452323896771647593, 5.91010760491596608798696119796, 6.25258896819233876180912111500, 6.51195174267065284423523367605, 6.60335267904141672083898668620
Plot not available for L-functions of degree greater than 10.
