| L(s) = 1 | − 3·5-s + 3·7-s + 4·11-s + 2·13-s − 3·17-s − 12·19-s + 25-s + 6·31-s − 9·35-s + 7·37-s − 6·41-s − 3·43-s + 9·47-s − 3·49-s − 18·53-s − 12·55-s − 12·59-s − 2·61-s − 6·65-s − 12·67-s − 9·71-s + 20·73-s + 12·77-s + 24·79-s − 10·83-s + 9·85-s + 6·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s + 1.20·11-s + 0.554·13-s − 0.727·17-s − 2.75·19-s + 1/5·25-s + 1.07·31-s − 1.52·35-s + 1.15·37-s − 0.937·41-s − 0.457·43-s + 1.31·47-s − 3/7·49-s − 2.47·53-s − 1.61·55-s − 1.56·59-s − 0.256·61-s − 0.744·65-s − 1.46·67-s − 1.06·71-s + 2.34·73-s + 1.36·77-s + 2.70·79-s − 1.09·83-s + 0.976·85-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72094241232081977650438220022, −7.70780230892393800680692613034, −6.84183841404954672785359877157, −6.61281408280301419881473003117, −6.41271512730125236255735154944, −6.26151843122698450260910629957, −5.59396507919488738155589650106, −5.18100658843356690000311929060, −4.54646553995832739439507953528, −4.47087145005024048048879342982, −4.17737711598404645504832062210, −4.02941976587680069192120515041, −3.29507607954844955381963802382, −3.13938350290223621990433141581, −2.19700950997448984117901349121, −2.18857996435631769302187881544, −1.36065443931923485091698259288, −1.21867796196456251070705972321, 0, 0,
1.21867796196456251070705972321, 1.36065443931923485091698259288, 2.18857996435631769302187881544, 2.19700950997448984117901349121, 3.13938350290223621990433141581, 3.29507607954844955381963802382, 4.02941976587680069192120515041, 4.17737711598404645504832062210, 4.47087145005024048048879342982, 4.54646553995832739439507953528, 5.18100658843356690000311929060, 5.59396507919488738155589650106, 6.26151843122698450260910629957, 6.41271512730125236255735154944, 6.61281408280301419881473003117, 6.84183841404954672785359877157, 7.70780230892393800680692613034, 7.72094241232081977650438220022
