| L(s) = 1 | − 4·3-s + 8·9-s + 8·13-s + 24·23-s − 2·25-s − 12·27-s + 24·37-s − 32·39-s − 8·47-s + 16·49-s + 16·59-s − 96·69-s + 16·71-s + 40·73-s + 8·75-s + 23·81-s + 8·83-s + 8·97-s + 56·107-s − 48·109-s − 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s + 2.21·13-s + 5.00·23-s − 2/5·25-s − 2.30·27-s + 3.94·37-s − 5.12·39-s − 1.16·47-s + 16/7·49-s + 2.08·59-s − 11.5·69-s + 1.89·71-s + 4.68·73-s + 0.923·75-s + 23/9·81-s + 0.878·83-s + 0.812·97-s + 5.41·107-s − 4.59·109-s − 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.648065654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648065654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4$ | \( ( 1 - 4 T - 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.07988302790223246056144120900, −6.89046630475133300534130116360, −6.63345930217755040312533339311, −6.51500579500628582876908205470, −6.23699734120006272994674980479, −6.03161736404331790213670155030, −5.85668004846700883932640289113, −5.54592286500107858042304277805, −5.51542667989407203918561645001, −5.01807871026151360302052969832, −4.85407820266115550208654748056, −4.81733059445413063157327450310, −4.73212765218850501635646726406, −3.94635010344571889670603757365, −3.85808596794281587373871446819, −3.83298226078509975248661279788, −3.32478347861792444160339932040, −3.15925824482276037665398143692, −2.64195822731942701521079276501, −2.31458868077247472819404367954, −2.19142868616185161522048972611, −1.32982430455142092471575026147, −0.889344124165243472400658211134, −0.865214052208346090838523450650, −0.822628267049516820040653816920,
0.822628267049516820040653816920, 0.865214052208346090838523450650, 0.889344124165243472400658211134, 1.32982430455142092471575026147, 2.19142868616185161522048972611, 2.31458868077247472819404367954, 2.64195822731942701521079276501, 3.15925824482276037665398143692, 3.32478347861792444160339932040, 3.83298226078509975248661279788, 3.85808596794281587373871446819, 3.94635010344571889670603757365, 4.73212765218850501635646726406, 4.81733059445413063157327450310, 4.85407820266115550208654748056, 5.01807871026151360302052969832, 5.51542667989407203918561645001, 5.54592286500107858042304277805, 5.85668004846700883932640289113, 6.03161736404331790213670155030, 6.23699734120006272994674980479, 6.51500579500628582876908205470, 6.63345930217755040312533339311, 6.89046630475133300534130116360, 7.07988302790223246056144120900
