| L(s) = 1 | + 2·4-s − 60·13-s − 26·19-s − 48·25-s + 6·31-s − 34·37-s − 340·43-s − 120·52-s − 144·61-s − 8·64-s − 86·67-s − 190·73-s − 52·76-s − 138·79-s − 64·97-s − 96·100-s − 122·103-s + 130·109-s − 192·121-s + 12·124-s + 127-s + 131-s + 137-s + 139-s − 68·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4.61·13-s − 1.36·19-s − 1.91·25-s + 6/31·31-s − 0.918·37-s − 7.90·43-s − 2.30·52-s − 2.36·61-s − 1/8·64-s − 1.28·67-s − 2.60·73-s − 0.684·76-s − 1.74·79-s − 0.659·97-s − 0.959·100-s − 1.18·103-s + 1.19·109-s − 1.58·121-s + 3/31·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.459·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001114584392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001114584392\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 48 T^{2} + 1679 T^{4} + 48 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 192 T^{2} + 22223 T^{4} + 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 450 T^{2} + 118979 T^{4} + 450 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 13 T - 192 T^{2} + 13 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 546 T^{2} + 18275 T^{4} + 546 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1170 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 3 T - 952 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 17 T - 1080 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3136 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 85 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 784 T^{2} - 4265025 T^{4} - 784 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4466 T^{2} + 12054675 T^{4} + 4466 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 1230 T^{2} - 10604461 T^{4} - 1230 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 72 T + 1463 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 43 T - 2640 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7344 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 95 T + 3696 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 69 T - 1480 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10080 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
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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
−6.96310006811695195403076663439, −6.96020437467112616414894780102, −6.89791882722364388931751900275, −6.41740064416713287894348508943, −6.09713155045370277675731228054, −6.06528913831099287802504792570, −5.68581790852428517463482733928, −5.20577298533688085537559027525, −5.04047642322960472550596782251, −5.03566780952908430956341602123, −4.90433779198642984932102237546, −4.44554467072740647459483146958, −4.36993682790679797714824942183, −3.89692384784599028842702539867, −3.76140648489348378440398762519, −3.15740220795478213568690229849, −2.97299494008525809463075989582, −2.84448165132518018074388291106, −2.62442256553562528986016741389, −2.04668376235492404444613472322, −1.82232010916850426049726352961, −1.69375424203999293160966340217, −1.51069089239873645298087727740, −0.12491806155648033746577812532, −0.02489218043299588480738138522,
0.02489218043299588480738138522, 0.12491806155648033746577812532, 1.51069089239873645298087727740, 1.69375424203999293160966340217, 1.82232010916850426049726352961, 2.04668376235492404444613472322, 2.62442256553562528986016741389, 2.84448165132518018074388291106, 2.97299494008525809463075989582, 3.15740220795478213568690229849, 3.76140648489348378440398762519, 3.89692384784599028842702539867, 4.36993682790679797714824942183, 4.44554467072740647459483146958, 4.90433779198642984932102237546, 5.03566780952908430956341602123, 5.04047642322960472550596782251, 5.20577298533688085537559027525, 5.68581790852428517463482733928, 6.06528913831099287802504792570, 6.09713155045370277675731228054, 6.41740064416713287894348508943, 6.89791882722364388931751900275, 6.96020437467112616414894780102, 6.96310006811695195403076663439
