| L(s) = 1 | − 2-s − 3·3-s + 4-s + 5-s + 3·6-s + 3·7-s + 8-s + 4·9-s − 10-s − 7·11-s − 3·12-s + 3·13-s − 3·14-s − 3·15-s − 3·16-s − 5·17-s − 4·18-s + 2·19-s + 20-s − 9·21-s + 7·22-s + 3·23-s − 3·24-s − 2·25-s − 3·26-s − 6·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 1.13·7-s + 0.353·8-s + 4/3·9-s − 0.316·10-s − 2.11·11-s − 0.866·12-s + 0.832·13-s − 0.801·14-s − 0.774·15-s − 3/4·16-s − 1.21·17-s − 0.942·18-s + 0.458·19-s + 0.223·20-s − 1.96·21-s + 1.49·22-s + 0.625·23-s − 0.612·24-s − 2/5·25-s − 0.588·26-s − 1.15·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123110 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123110 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 947 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 38 T + p T^{2} ) \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T - 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 147 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 129 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 99 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 180 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−13.8340513103, −13.5584358203, −13.1551070406, −12.8196790815, −12.1847348522, −11.6350306531, −11.1720780854, −11.0048196387, −10.8136528283, −10.3637141800, −9.78162166423, −9.25997362823, −8.66269635462, −8.08918023406, −7.66257928211, −7.35031807476, −6.62439039264, −6.02717804044, −5.69002962240, −5.13495127562, −4.73022158723, −4.19725632493, −2.86597294960, −2.11965863304, −1.29651221043, 0,
1.29651221043, 2.11965863304, 2.86597294960, 4.19725632493, 4.73022158723, 5.13495127562, 5.69002962240, 6.02717804044, 6.62439039264, 7.35031807476, 7.66257928211, 8.08918023406, 8.66269635462, 9.25997362823, 9.78162166423, 10.3637141800, 10.8136528283, 11.0048196387, 11.1720780854, 11.6350306531, 12.1847348522, 12.8196790815, 13.1551070406, 13.5584358203, 13.8340513103
