L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s − 4·10-s + 6·12-s + 4·14-s + 4·15-s + 5·16-s − 8·17-s − 6·18-s + 6·19-s + 6·20-s − 4·21-s − 8·23-s − 8·24-s − 4·25-s + 4·27-s − 6·28-s − 14·29-s − 8·30-s + 10·31-s − 6·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.73·12-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 1.37·19-s + 1.34·20-s − 0.872·21-s − 1.66·23-s − 1.63·24-s − 4/5·25-s + 0.769·27-s − 1.13·28-s − 2.59·29-s − 1.46·30-s + 1.79·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 95 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 95 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 284 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 248 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.79595565510396281140199756205, −7.74755979925477219971897586483, −6.97941495296668644432644280496, −6.96762544505450822739438257732, −6.35236701969356359766549416081, −6.31561391025388964703297248124, −5.80034976908050005797436296857, −5.50612516894817533987685864571, −4.97770222669160003935369140363, −4.33311834499681691549442479500, −3.99913991592907148968147221830, −3.73619262539710172944950808167, −3.08765140932561698917316733758, −2.72987177916365027057390265447, −2.34922486434686809704694169150, −2.12704623228384191663301870260, −1.42467150018880980312904841079, −1.33033726759727835316665417796, 0, 0,
1.33033726759727835316665417796, 1.42467150018880980312904841079, 2.12704623228384191663301870260, 2.34922486434686809704694169150, 2.72987177916365027057390265447, 3.08765140932561698917316733758, 3.73619262539710172944950808167, 3.99913991592907148968147221830, 4.33311834499681691549442479500, 4.97770222669160003935369140363, 5.50612516894817533987685864571, 5.80034976908050005797436296857, 6.31561391025388964703297248124, 6.35236701969356359766549416081, 6.96762544505450822739438257732, 6.97941495296668644432644280496, 7.74755979925477219971897586483, 7.79595565510396281140199756205