| L(s) = 1 | + 3·4-s + 18·7-s − 6·13-s − 3·16-s − 18·19-s − 3·25-s + 54·28-s + 36·31-s − 6·37-s + 12·43-s + 153·49-s − 18·52-s + 30·61-s − 24·64-s − 6·73-s − 54·76-s + 42·79-s − 108·91-s − 6·97-s − 9·100-s + 24·103-s − 30·109-s − 54·112-s − 21·121-s + 108·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 6.80·7-s − 1.66·13-s − 3/4·16-s − 4.12·19-s − 3/5·25-s + 10.2·28-s + 6.46·31-s − 0.986·37-s + 1.82·43-s + 21.8·49-s − 2.49·52-s + 3.84·61-s − 3·64-s − 0.702·73-s − 6.19·76-s + 4.72·79-s − 11.3·91-s − 0.609·97-s − 0.899·100-s + 2.36·103-s − 2.87·109-s − 5.10·112-s − 1.90·121-s + 9.69·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(41.15737317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(41.15737317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} + 3 p^{2} T^{4} - 21 T^{6} + 3 p^{4} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 3 T^{2} + 21 T^{4} + 201 T^{6} + 21 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 9 T + 45 T^{2} - 145 T^{3} + 45 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 + 21 T^{2} + 21 p T^{4} + 1681 T^{6} + 21 p^{3} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 21 T^{2} + 41 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 + 9 T + 63 T^{2} + 289 T^{3} + 63 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 27 T^{2} + 1551 T^{4} + 27765 T^{6} + 1551 p^{2} T^{8} + 27 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 102 T^{2} + 5802 T^{4} + 204901 T^{6} + 5802 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 18 T + 198 T^{2} - 1315 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 3 T + 93 T^{2} + 239 T^{3} + 93 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 84 T^{2} + 4362 T^{4} + 212137 T^{6} + 4362 p^{2} T^{8} + 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 6 T + 3 p T^{2} - 508 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 174 T^{2} + 13794 T^{4} + 734569 T^{6} + 13794 p^{2} T^{8} + 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 138 T^{2} + 12156 T^{4} + 802177 T^{6} + 12156 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 + 93 T^{2} + 7467 T^{4} + 635569 T^{6} + 7467 p^{2} T^{8} + 93 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 15 T + 249 T^{2} - 1901 T^{3} + 249 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 + 117 T^{2} + 296 T^{3} + 117 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 + 255 T^{2} + 36066 T^{4} + 3117063 T^{6} + 36066 p^{2} T^{8} + 255 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 3 T + 3 p T^{2} + 435 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - 21 T + 372 T^{2} - 3585 T^{3} + 372 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 150 T^{2} + 17130 T^{4} + 1339593 T^{6} + 17130 p^{2} T^{8} + 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 378 T^{2} + 67548 T^{4} + 7413705 T^{6} + 67548 p^{2} T^{8} + 378 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 3 T + 186 T^{2} + 691 T^{3} + 186 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.71793018035880546574441844242, −4.46918886727703756125996093325, −4.37057117085401781746085675748, −4.23021736437954355064180205222, −4.16302824421255798477156365821, −4.07076114166919703304967736213, −4.01895190448000277138250741969, −3.58082834864999070770811189724, −3.53802888330195643524042102655, −2.98385328248569188320135099233, −2.84185667338419255377471277346, −2.66236431763781340682856981832, −2.45690048919528752554738086493, −2.45020536397727552541107616173, −2.44243810478537382315465210172, −2.21644406313980669069379853385, −1.93179692381627343983668343666, −1.86503508468010769653005053569, −1.75923411543440384550294682894, −1.65942147574243082092831241136, −1.49055460234648828271613558203, −0.991166617242083776648513697357, −0.969255112279974773142271910569, −0.64488747070857327521882211635, −0.44198945792362477908313222150,
0.44198945792362477908313222150, 0.64488747070857327521882211635, 0.969255112279974773142271910569, 0.991166617242083776648513697357, 1.49055460234648828271613558203, 1.65942147574243082092831241136, 1.75923411543440384550294682894, 1.86503508468010769653005053569, 1.93179692381627343983668343666, 2.21644406313980669069379853385, 2.44243810478537382315465210172, 2.45020536397727552541107616173, 2.45690048919528752554738086493, 2.66236431763781340682856981832, 2.84185667338419255377471277346, 2.98385328248569188320135099233, 3.53802888330195643524042102655, 3.58082834864999070770811189724, 4.01895190448000277138250741969, 4.07076114166919703304967736213, 4.16302824421255798477156365821, 4.23021736437954355064180205222, 4.37057117085401781746085675748, 4.46918886727703756125996093325, 4.71793018035880546574441844242
Plot not available for L-functions of degree greater than 10.
