| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 4·5-s + 6·6-s − 7-s − 3·8-s + 9-s + 12·10-s − 8·12-s − 13-s + 3·14-s + 8·15-s + 3·16-s − 3·18-s − 4·19-s − 16·20-s + 2·21-s + 2·23-s + 6·24-s + 5·25-s + 3·26-s − 2·27-s − 4·28-s − 24·30-s + 4·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 2.44·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 3.79·10-s − 2.30·12-s − 0.277·13-s + 0.801·14-s + 2.06·15-s + 3/4·16-s − 0.707·18-s − 0.917·19-s − 3.57·20-s + 0.436·21-s + 0.417·23-s + 1.22·24-s + 25-s + 0.588·26-s − 0.384·27-s − 0.755·28-s − 4.38·30-s + 0.718·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 177 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T - 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 121 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.4901123623, −19.1188559049, −18.9941196049, −18.1962238572, −17.5803652188, −17.1605094556, −16.8851484436, −16.0601464230, −15.7560683073, −15.1661355976, −14.3513730910, −13.1074126825, −12.3110918435, −11.8701368771, −11.2395985780, −10.7353474786, −10.0294591025, −9.31948512140, −8.38401905615, −8.11597358074, −7.20784764151, −6.47935710087, −5.14407534839, −3.75241654741, 0,
3.75241654741, 5.14407534839, 6.47935710087, 7.20784764151, 8.11597358074, 8.38401905615, 9.31948512140, 10.0294591025, 10.7353474786, 11.2395985780, 11.8701368771, 12.3110918435, 13.1074126825, 14.3513730910, 15.1661355976, 15.7560683073, 16.0601464230, 16.8851484436, 17.1605094556, 17.5803652188, 18.1962238572, 18.9941196049, 19.1188559049, 19.4901123623
