| L(s) = 1 | − 4·11-s + 2·13-s − 4·17-s + 8·23-s − 2·25-s + 4·29-s − 8·31-s + 4·37-s − 16·41-s − 8·43-s + 12·47-s − 6·49-s − 4·53-s + 4·59-s − 4·61-s − 8·67-s − 4·71-s + 12·73-s − 4·83-s − 24·89-s − 4·97-s + 4·101-s + 16·103-s + 12·109-s − 12·113-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s − 0.970·17-s + 1.66·23-s − 2/5·25-s + 0.742·29-s − 1.43·31-s + 0.657·37-s − 2.49·41-s − 1.21·43-s + 1.75·47-s − 6/7·49-s − 0.549·53-s + 0.520·59-s − 0.512·61-s − 0.977·67-s − 0.474·71-s + 1.40·73-s − 0.439·83-s − 2.54·89-s − 0.406·97-s + 0.398·101-s + 1.57·103-s + 1.14·109-s − 1.12·113-s − 0.909·121-s + 0.0887·127-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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.50882219033294999054270675561, −7.42288805120005157617126359742, −7.14074172420249309446950759965, −6.63568150208593029229226485823, −6.27352120781029551437112790242, −6.18148805351468343467770220224, −5.36716580915912102267263174021, −5.26479690119313308466315817811, −5.05836116475561885098897298631, −4.53366358269159598803429423439, −4.18216979925816024728204923933, −3.69289474811628930494470789311, −3.13950386261738642154802044268, −3.13304087856937316966797121124, −2.31789467907068771219507706263, −2.24774636445659575553930432772, −1.40056564454665554499467224673, −1.17893366823459620158524090909, 0, 0,
1.17893366823459620158524090909, 1.40056564454665554499467224673, 2.24774636445659575553930432772, 2.31789467907068771219507706263, 3.13304087856937316966797121124, 3.13950386261738642154802044268, 3.69289474811628930494470789311, 4.18216979925816024728204923933, 4.53366358269159598803429423439, 5.05836116475561885098897298631, 5.26479690119313308466315817811, 5.36716580915912102267263174021, 6.18148805351468343467770220224, 6.27352120781029551437112790242, 6.63568150208593029229226485823, 7.14074172420249309446950759965, 7.42288805120005157617126359742, 7.50882219033294999054270675561
