| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 4·5-s + 8·6-s − 5·7-s + 2·8-s + 7·9-s + 8·10-s − 4·12-s − 6·13-s + 10·14-s + 16·15-s − 3·16-s − 17-s − 14·18-s + 2·19-s − 4·20-s + 20·21-s − 3·23-s − 8·24-s + 3·25-s + 12·26-s − 4·27-s − 5·28-s − 32·30-s − 2·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 1.88·7-s + 0.707·8-s + 7/3·9-s + 2.52·10-s − 1.15·12-s − 1.66·13-s + 2.67·14-s + 4.13·15-s − 3/4·16-s − 0.242·17-s − 3.29·18-s + 0.458·19-s − 0.894·20-s + 4.36·21-s − 0.625·23-s − 1.63·24-s + 3/5·25-s + 2.35·26-s − 0.769·27-s − 0.944·28-s − 5.84·30-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5414 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5414 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 2707 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 40 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 7 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 69 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T - 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 46 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 143 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 248 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 137 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−17.5959181428, −17.4415998362, −17.0339368008, −16.4211771454, −16.0839778255, −15.9653145027, −15.3007131255, −14.5264996437, −13.5862332275, −12.7979561829, −12.3636705046, −11.8537502773, −11.7220289885, −10.9679444152, −10.2425126027, −10.0976434598, −9.37651295116, −8.69600717630, −7.69114820022, −7.38394311286, −6.69783680029, −6.02145125686, −5.17105980310, −4.39446486011, −3.36113545819, 0, 0,
3.36113545819, 4.39446486011, 5.17105980310, 6.02145125686, 6.69783680029, 7.38394311286, 7.69114820022, 8.69600717630, 9.37651295116, 10.0976434598, 10.2425126027, 10.9679444152, 11.7220289885, 11.8537502773, 12.3636705046, 12.7979561829, 13.5862332275, 14.5264996437, 15.3007131255, 15.9653145027, 16.0839778255, 16.4211771454, 17.0339368008, 17.4415998362, 17.5959181428
