| L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s − 12·17-s + 8·19-s − 6·25-s − 4·27-s + 16·33-s + 4·41-s − 8·43-s + 2·49-s + 24·51-s − 16·57-s + 8·59-s − 8·67-s − 12·73-s + 12·75-s + 5·81-s − 24·83-s + 20·89-s − 28·97-s − 24·99-s + 8·107-s + 4·113-s + 26·121-s − 8·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s − 2.91·17-s + 1.83·19-s − 6/5·25-s − 0.769·27-s + 2.78·33-s + 0.624·41-s − 1.21·43-s + 2/7·49-s + 3.36·51-s − 2.11·57-s + 1.04·59-s − 0.977·67-s − 1.40·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 2.11·89-s − 2.84·97-s − 2.41·99-s + 0.773·107-s + 0.376·113-s + 2.36·121-s − 0.721·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.84849442743057013529606410604, −10.20586918892848667331432436971, −9.826637660192491869357018654726, −9.133901360741508328144538474134, −8.391046533691483698004382219288, −7.83182394757322498939219938639, −7.17212008872630953090710242182, −6.80946897903754876349072202165, −5.77092473843782707280190480124, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.31158789218116792910899233548, −3.00571163091752467289605693825, −2.09803110754033007053716721218, 0,
2.09803110754033007053716721218, 3.00571163091752467289605693825, 4.31158789218116792910899233548, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 5.77092473843782707280190480124, 6.80946897903754876349072202165, 7.17212008872630953090710242182, 7.83182394757322498939219938639, 8.391046533691483698004382219288, 9.133901360741508328144538474134, 9.826637660192491869357018654726, 10.20586918892848667331432436971, 10.84849442743057013529606410604
