| L(s) = 1 | + 12·7-s − 20·13-s + 8·19-s − 18·25-s + 44·31-s − 24·37-s − 164·43-s + 134·49-s + 172·61-s − 4·67-s + 328·73-s − 20·79-s − 240·91-s + 188·97-s + 268·103-s + 40·109-s − 210·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 12/7·7-s − 1.53·13-s + 8/19·19-s − 0.719·25-s + 1.41·31-s − 0.648·37-s − 3.81·43-s + 2.73·49-s + 2.81·61-s − 0.0597·67-s + 4.49·73-s − 0.253·79-s − 2.63·91-s + 1.93·97-s + 2.60·103-s + 0.366·109-s − 1.73·121-s + 0.00787·127-s + 0.00763·131-s + 0.721·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.323022622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.323022622\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 18 T^{2} - 301 T^{4} + 18 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T - 13 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 210 T^{2} + 29459 T^{4} + 210 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 10 T - 69 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 930 T^{2} + 585059 T^{4} + 930 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1394 T^{2} + 1235955 T^{4} + 1394 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 22 T - 477 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 2210 T^{2} + 2058339 T^{4} + 2210 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 82 T + 4875 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 190 T^{2} - 4843581 T^{4} - 190 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 1746 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 1554 T^{2} - 9702445 T^{4} + 1554 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T + 3675 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 4485 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5406 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 10 T - 6141 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 8370 T^{2} + 22598579 T^{4} + 8370 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 14690 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.57143116983371955603583918920, −7.15503848744656392173095029078, −6.92673510746324879163474948228, −6.72858708454849368259536977374, −6.50152581484569674922836621208, −6.22670716344817624826961666046, −6.03922595575619394758102089440, −5.38258901668566012169923563280, −5.32754310119831671937951955415, −5.31401405845741258720619847810, −4.84911888966896128966657426662, −4.82089399783545044689052923493, −4.65167888076932681093142320981, −4.06737185896976583603817403615, −3.96626244102497082618344059082, −3.57617183181257244435044487522, −3.33960698552494927329551399692, −2.99705130277269218367305233527, −2.54245518389451309474752685681, −2.17812748699131714504723621781, −2.01251257145368673643389970499, −1.81495839391927337244744552181, −1.24308990849114479282150036838, −0.65917763751902512153555662181, −0.50804383660659676917576369198,
0.50804383660659676917576369198, 0.65917763751902512153555662181, 1.24308990849114479282150036838, 1.81495839391927337244744552181, 2.01251257145368673643389970499, 2.17812748699131714504723621781, 2.54245518389451309474752685681, 2.99705130277269218367305233527, 3.33960698552494927329551399692, 3.57617183181257244435044487522, 3.96626244102497082618344059082, 4.06737185896976583603817403615, 4.65167888076932681093142320981, 4.82089399783545044689052923493, 4.84911888966896128966657426662, 5.31401405845741258720619847810, 5.32754310119831671937951955415, 5.38258901668566012169923563280, 6.03922595575619394758102089440, 6.22670716344817624826961666046, 6.50152581484569674922836621208, 6.72858708454849368259536977374, 6.92673510746324879163474948228, 7.15503848744656392173095029078, 7.57143116983371955603583918920
