L(s) = 1 | − 2·3-s + 3·9-s + 2·13-s + 2·17-s + 4·23-s − 4·27-s − 6·29-s − 8·31-s − 8·37-s − 4·39-s − 12·41-s + 4·43-s + 4·47-s − 11·49-s − 4·51-s − 10·53-s − 18·61-s + 12·67-s − 8·69-s − 8·71-s + 28·79-s + 5·81-s + 24·83-s + 12·87-s + 8·89-s + 16·93-s − 4·97-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 0.554·13-s + 0.485·17-s + 0.834·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.31·37-s − 0.640·39-s − 1.87·41-s + 0.609·43-s + 0.583·47-s − 1.57·49-s − 0.560·51-s − 1.37·53-s − 2.30·61-s + 1.46·67-s − 0.963·69-s − 0.949·71-s + 3.15·79-s + 5/9·81-s + 2.63·83-s + 1.28·87-s + 0.847·89-s + 1.65·93-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 83 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 191 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 167 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 307 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 146 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.52846404653819358300152975543, −7.46911026185656772871379418515, −6.79711989034078450914933996567, −6.62905062643755033230210897924, −6.26769544841163805648943098354, −6.09505813854917591311812753508, −5.35686278140064492046068272778, −5.31491076838419690578177269878, −5.01989992191009929941349411433, −4.70025430564873472196244825207, −4.09700886591462600377067140390, −3.62723902045274175384313069552, −3.41146935699955689089799673479, −3.17881188964903888592991475725, −2.18451649306526216421977700908, −2.02799230339822298338267029124, −1.30774773316930170918581049325, −1.13396214535393606718377049988, 0, 0,
1.13396214535393606718377049988, 1.30774773316930170918581049325, 2.02799230339822298338267029124, 2.18451649306526216421977700908, 3.17881188964903888592991475725, 3.41146935699955689089799673479, 3.62723902045274175384313069552, 4.09700886591462600377067140390, 4.70025430564873472196244825207, 5.01989992191009929941349411433, 5.31491076838419690578177269878, 5.35686278140064492046068272778, 6.09505813854917591311812753508, 6.26769544841163805648943098354, 6.62905062643755033230210897924, 6.79711989034078450914933996567, 7.46911026185656772871379418515, 7.52846404653819358300152975543