| L(s) = 1 | − 2-s − 4·3-s + 4-s − 4·5-s + 4·6-s − 5·7-s − 8-s + 7·9-s + 4·10-s − 3·11-s − 4·12-s − 2·13-s + 5·14-s + 16·15-s + 16-s − 7·18-s − 5·19-s − 4·20-s + 20·21-s + 3·22-s − 23-s + 4·24-s + 5·25-s + 2·26-s − 4·27-s − 5·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s − 1.88·7-s − 0.353·8-s + 7/3·9-s + 1.26·10-s − 0.904·11-s − 1.15·12-s − 0.554·13-s + 1.33·14-s + 4.13·15-s + 1/4·16-s − 1.64·18-s − 1.14·19-s − 0.894·20-s + 4.36·21-s + 0.639·22-s − 0.208·23-s + 0.816·24-s + 25-s + 0.392·26-s − 0.769·27-s − 0.944·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 1627 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 49 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 11 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 31 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 73 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 84 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 77 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 95 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−16.5186058696, −16.3925937828, −15.9972716123, −15.7634778999, −15.0371578983, −14.7763269370, −13.4801207566, −12.9307833076, −12.5543401972, −12.1526146014, −11.6389104909, −11.3867880811, −10.7423194628, −10.3560098681, −9.91998080700, −9.14489297321, −8.37725416387, −7.70253375698, −7.14359633871, −6.59735217556, −6.07009308089, −5.51986781606, −4.63297227372, −3.84463859233, −2.87815990777, 0, 0,
2.87815990777, 3.84463859233, 4.63297227372, 5.51986781606, 6.07009308089, 6.59735217556, 7.14359633871, 7.70253375698, 8.37725416387, 9.14489297321, 9.91998080700, 10.3560098681, 10.7423194628, 11.3867880811, 11.6389104909, 12.1526146014, 12.5543401972, 12.9307833076, 13.4801207566, 14.7763269370, 15.0371578983, 15.7634778999, 15.9972716123, 16.3925937828, 16.5186058696
