| L(s) = 1 | + 2·4-s − 2·9-s − 4·11-s + 3·16-s − 20·19-s − 16·29-s − 4·36-s − 4·41-s − 8·44-s + 22·49-s − 44·59-s + 24·61-s + 12·64-s + 20·71-s − 40·76-s − 36·79-s + 3·81-s + 8·89-s + 8·99-s + 20·101-s + 24·109-s − 32·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s − 2/3·9-s − 1.20·11-s + 3/4·16-s − 4.58·19-s − 2.97·29-s − 2/3·36-s − 0.624·41-s − 1.20·44-s + 22/7·49-s − 5.72·59-s + 3.07·61-s + 3/2·64-s + 2.37·71-s − 4.58·76-s − 4.05·79-s + 1/3·81-s + 0.847·89-s + 0.804·99-s + 1.99·101-s + 2.29·109-s − 2.97·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5946777740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5946777740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1531 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 114 T^{2} + 5699 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 36 T^{2} - 586 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 42 T^{2} + 4571 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8726 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 159 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19910 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 15254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 354 T^{2} + 49859 T^{4} - 354 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.20947480244239138568305514883, −7.17083799543508625832577708136, −6.93892307643493041211638336217, −6.60048927204186084311149702385, −6.50340754338478934264957601481, −6.10750931422260296497565758890, −5.83852154358371037655092103998, −5.80985558517612891417254315971, −5.71912346303800481733446539975, −5.36660791819723302090138795322, −4.91288426719213226642582124154, −4.77461277117628869756513046157, −4.38292862876907506593037439814, −4.22333671755731136566954811878, −3.86874698775196823738142436214, −3.85985158621191175219979699652, −3.20427831868013738321927199387, −3.01589872454623983117308675566, −2.90379806582862881081652373243, −2.19215353786829350359294181795, −2.03340319150489281258903234508, −1.97926463275592165519936332302, −1.92627424087933218866930952165, −0.835537525156124629741059187479, −0.20704600420972331465068024152,
0.20704600420972331465068024152, 0.835537525156124629741059187479, 1.92627424087933218866930952165, 1.97926463275592165519936332302, 2.03340319150489281258903234508, 2.19215353786829350359294181795, 2.90379806582862881081652373243, 3.01589872454623983117308675566, 3.20427831868013738321927199387, 3.85985158621191175219979699652, 3.86874698775196823738142436214, 4.22333671755731136566954811878, 4.38292862876907506593037439814, 4.77461277117628869756513046157, 4.91288426719213226642582124154, 5.36660791819723302090138795322, 5.71912346303800481733446539975, 5.80985558517612891417254315971, 5.83852154358371037655092103998, 6.10750931422260296497565758890, 6.50340754338478934264957601481, 6.60048927204186084311149702385, 6.93892307643493041211638336217, 7.17083799543508625832577708136, 7.20947480244239138568305514883
