| L(s) = 1 | + 3-s + 4-s − 8·7-s + 9-s + 12-s − 8·13-s + 16-s + 2·19-s − 8·21-s − 10·25-s + 27-s − 8·28-s + 4·31-s + 36-s − 8·37-s − 8·39-s − 8·43-s + 48-s + 34·49-s − 8·52-s + 2·57-s + 28·61-s − 8·63-s + 64-s + 16·67-s + 28·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 0.458·19-s − 1.74·21-s − 2·25-s + 0.192·27-s − 1.51·28-s + 0.718·31-s + 1/6·36-s − 1.31·37-s − 1.28·39-s − 1.21·43-s + 0.144·48-s + 34/7·49-s − 1.10·52-s + 0.264·57-s + 3.58·61-s − 1.00·63-s + 1/8·64-s + 1.95·67-s + 3.27·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38988 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38988 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.814927382770120407835166865620, −9.659160529040683538070775225454, −9.465004478872581430365849790992, −8.381778434403041270610948676969, −7.973335490421157373200340268575, −7.06756557503064388001592546481, −6.84732233829542762166868447522, −6.59547283064248335138694127148, −5.61719283222003359694793802526, −5.21083603966480863402338802043, −3.89674503164637365373404147869, −3.58677867797817326821260752741, −2.72251484544452322835217288408, −2.34092707995523044468378525917, 0,
2.34092707995523044468378525917, 2.72251484544452322835217288408, 3.58677867797817326821260752741, 3.89674503164637365373404147869, 5.21083603966480863402338802043, 5.61719283222003359694793802526, 6.59547283064248335138694127148, 6.84732233829542762166868447522, 7.06756557503064388001592546481, 7.973335490421157373200340268575, 8.381778434403041270610948676969, 9.465004478872581430365849790992, 9.659160529040683538070775225454, 9.814927382770120407835166865620
