| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s − 6·7-s − 2·8-s + 9-s − 6·11-s + 2·12-s − 12·14-s − 4·16-s + 2·18-s − 6·19-s − 12·21-s − 12·22-s + 6·23-s − 4·24-s − 14·25-s − 2·27-s − 6·28-s + 6·29-s + 24·31-s − 2·32-s − 12·33-s + 36-s − 6·37-s − 12·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s − 2.26·7-s − 0.707·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 3.20·14-s − 16-s + 0.471·18-s − 1.37·19-s − 2.61·21-s − 2.55·22-s + 1.25·23-s − 0.816·24-s − 2.79·25-s − 0.384·27-s − 1.13·28-s + 1.11·29-s + 4.31·31-s − 0.353·32-s − 2.08·33-s + 1/6·36-s − 0.986·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3489397071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3489397071\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 16 T^{2} + 36 T^{3} + 99 T^{4} + 36 p T^{5} + 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 8 T^{2} + 108 T^{3} - 573 T^{4} + 108 p T^{5} + 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 43 T^{2} - 18 T^{3} + 3084 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 52 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 74 T^{2} + 1995 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 112 T^{2} + 1404 T^{3} + 18747 T^{4} + 1404 p T^{5} + 112 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 88 T^{2} - 108 T^{3} + 7779 T^{4} - 108 p T^{5} - 88 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 58 T^{2} + 288 T^{3} + 20067 T^{4} + 288 p T^{5} - 58 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.03740589894413140481202814916, −6.77093812575731718446507195545, −6.69147128074287386935343135040, −6.46063652865219640565881988374, −6.01049729827333751081862113587, −5.99590240773074487576059578271, −5.91227576191916703090274079315, −5.56907895733564438168329298476, −5.28060887590775151590693126117, −5.07542507110338206978610587255, −4.58311455789743975125895536488, −4.47509796032712058188361522757, −4.37230331736946274036767094605, −4.12825681442363055605613855590, −3.89242170164298322800130144199, −3.20852158727551795985948368379, −3.19013151598015824790781444438, −3.05955010290036586529054931564, −2.87091240016748578150206092308, −2.82146694988169282767981018446, −2.28671665363405871728820345908, −1.94108609396576904762495473164, −1.62411912432382812906371963691, −0.77903028533538935146245994635, −0.11122698193967864768077445363,
0.11122698193967864768077445363, 0.77903028533538935146245994635, 1.62411912432382812906371963691, 1.94108609396576904762495473164, 2.28671665363405871728820345908, 2.82146694988169282767981018446, 2.87091240016748578150206092308, 3.05955010290036586529054931564, 3.19013151598015824790781444438, 3.20852158727551795985948368379, 3.89242170164298322800130144199, 4.12825681442363055605613855590, 4.37230331736946274036767094605, 4.47509796032712058188361522757, 4.58311455789743975125895536488, 5.07542507110338206978610587255, 5.28060887590775151590693126117, 5.56907895733564438168329298476, 5.91227576191916703090274079315, 5.99590240773074487576059578271, 6.01049729827333751081862113587, 6.46063652865219640565881988374, 6.69147128074287386935343135040, 6.77093812575731718446507195545, 7.03740589894413140481202814916
