| L(s) = 1 | − 4·2-s + 2·4-s + 14·8-s − 2·11-s − 21·16-s + 8·22-s − 18·23-s − 6·25-s + 2·29-s − 14·32-s − 10·37-s − 30·43-s − 4·44-s + 72·46-s + 24·50-s − 30·53-s − 8·58-s + 35·64-s + 30·67-s + 2·71-s + 40·74-s − 10·79-s + 120·86-s − 28·88-s − 36·92-s − 12·100-s + 120·106-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s + 4.94·8-s − 0.603·11-s − 5.25·16-s + 1.70·22-s − 3.75·23-s − 6/5·25-s + 0.371·29-s − 2.47·32-s − 1.64·37-s − 4.57·43-s − 0.603·44-s + 10.6·46-s + 3.39·50-s − 4.12·53-s − 1.05·58-s + 35/8·64-s + 3.66·67-s + 0.237·71-s + 4.64·74-s − 1.12·79-s + 12.9·86-s − 2.98·88-s − 3.75·92-s − 6/5·100-s + 11.6·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 6 T^{2} + 59 T^{4} + 308 T^{6} + 59 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + 31 T^{2} + 21 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 8 T^{2} + 463 T^{4} + 2360 T^{6} + 463 p^{2} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 54 T^{2} + 1643 T^{4} + 34244 T^{6} + 1643 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 4 p T^{2} + 2747 T^{4} + 63000 T^{6} + 2747 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 9 T + 89 T^{2} + 413 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - T + 71 T^{2} - p T^{3} + 71 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 82 T^{2} + 5059 T^{4} + 176428 T^{6} + 5059 p^{2} T^{8} + 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 5 T + 89 T^{2} + 357 T^{3} + 89 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 134 T^{2} + 9003 T^{4} + 414212 T^{6} + 9003 p^{2} T^{8} + 134 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 15 T + 155 T^{2} + 1121 T^{3} + 155 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 34 T^{2} + 675 T^{4} + 81676 T^{6} + 675 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 15 T + 213 T^{2} + 1617 T^{3} + 213 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 4 p T^{2} + 28551 T^{4} + 2097536 T^{6} + 28551 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 188 T^{2} + 12799 T^{4} + 616112 T^{6} + 12799 p^{2} T^{8} + 188 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 15 T + 227 T^{2} - 1939 T^{3} + 227 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - T + 197 T^{2} - 113 T^{3} + 197 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 4 p T^{2} + 40423 T^{4} + 3567424 T^{6} + 40423 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 5 T + 131 T^{2} + 987 T^{3} + 131 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 428 T^{2} + 81663 T^{4} + 8791280 T^{6} + 81663 p^{2} T^{8} + 428 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 300 T^{2} + 48471 T^{4} + 5116048 T^{6} + 48471 p^{2} T^{8} + 300 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 190 T^{2} + 17851 T^{4} + 1155988 T^{6} + 17851 p^{2} T^{8} + 190 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.12618865264330243272586809562, −4.66242706269522694305827709496, −4.64906436221182076322044504788, −4.47915830199782476805293226023, −4.44537255823646152222658773937, −4.12648640708706317864269990440, −4.10740368848145110878959535987, −3.78279288461116436784027301403, −3.65368739731125816427269044441, −3.62401548032912201393056299150, −3.60213056835530708345226261343, −3.57622705351919046790424166096, −3.11776995211817238943642268717, −3.01121449897725802775781667537, −2.65312714815011980343580339543, −2.52928602321622158164488801854, −2.41301412003736866739526543226, −2.24883966860982533971088291485, −2.13550346477135690125846673432, −1.77059314042873682099767305519, −1.48326909289241056743173883750, −1.39141912135037187746182769772, −1.28845824247217447805538450026, −1.16579496479026867121916089849, −1.06158133095603163744717622414, 0, 0, 0, 0, 0, 0,
1.06158133095603163744717622414, 1.16579496479026867121916089849, 1.28845824247217447805538450026, 1.39141912135037187746182769772, 1.48326909289241056743173883750, 1.77059314042873682099767305519, 2.13550346477135690125846673432, 2.24883966860982533971088291485, 2.41301412003736866739526543226, 2.52928602321622158164488801854, 2.65312714815011980343580339543, 3.01121449897725802775781667537, 3.11776995211817238943642268717, 3.57622705351919046790424166096, 3.60213056835530708345226261343, 3.62401548032912201393056299150, 3.65368739731125816427269044441, 3.78279288461116436784027301403, 4.10740368848145110878959535987, 4.12648640708706317864269990440, 4.44537255823646152222658773937, 4.47915830199782476805293226023, 4.64906436221182076322044504788, 4.66242706269522694305827709496, 5.12618865264330243272586809562
Plot not available for L-functions of degree greater than 10.
