| L(s) = 1 | − 2·2-s − 3-s + 4-s − 4·5-s + 2·6-s − 3·7-s + 2·8-s + 8·10-s − 2·11-s − 12-s + 6·14-s + 4·15-s − 3·16-s − 3·19-s − 4·20-s + 3·21-s + 4·22-s + 4·23-s − 2·24-s + 6·25-s + 27-s − 3·28-s + 4·29-s − 8·30-s − 8·31-s − 2·32-s + 2·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.13·7-s + 0.707·8-s + 2.52·10-s − 0.603·11-s − 0.288·12-s + 1.60·14-s + 1.03·15-s − 3/4·16-s − 0.688·19-s − 0.894·20-s + 0.654·21-s + 0.852·22-s + 0.834·23-s − 0.408·24-s + 6/5·25-s + 0.192·27-s − 0.566·28-s + 0.742·29-s − 1.46·30-s − 1.43·31-s − 0.353·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1854 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1854 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 4 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T - 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 97 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 97 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 112 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.0399359548, −18.8462348521, −18.3383283526, −17.6061170897, −17.1725811728, −16.4928878546, −16.1972660211, −15.7795526372, −15.1217705071, −14.5793329658, −13.4666386829, −12.9196935624, −12.3872928001, −11.7186207500, −11.0566599756, −10.5865773262, −9.93853914509, −9.20396812780, −8.50598385584, −8.02017188282, −7.25165585403, −6.70123933152, −5.43396274455, −4.32565511863, −3.32257348942, 0,
3.32257348942, 4.32565511863, 5.43396274455, 6.70123933152, 7.25165585403, 8.02017188282, 8.50598385584, 9.20396812780, 9.93853914509, 10.5865773262, 11.0566599756, 11.7186207500, 12.3872928001, 12.9196935624, 13.4666386829, 14.5793329658, 15.1217705071, 15.7795526372, 16.1972660211, 16.4928878546, 17.1725811728, 17.6061170897, 18.3383283526, 18.8462348521, 19.0399359548
