| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 3·7-s + 8-s − 10-s − 11-s − 2·13-s − 3·14-s + 15-s − 16-s − 6·19-s + 3·21-s − 22-s − 2·23-s − 24-s − 2·26-s + 4·27-s + 30-s − 7·31-s − 6·32-s + 33-s + 3·35-s + 2·37-s − 6·38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s − 0.801·14-s + 0.258·15-s − 1/4·16-s − 1.37·19-s + 0.654·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 0.392·26-s + 0.769·27-s + 0.182·30-s − 1.25·31-s − 1.06·32-s + 0.174·33-s + 0.507·35-s + 0.328·37-s − 0.973·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 10163 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 54 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 195 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−15.6493611928, −14.9805811271, −14.5416374046, −14.0593119955, −13.6838905933, −12.8832229728, −12.6834405920, −12.5875643686, −11.8425963982, −11.1951872686, −10.8102860044, −10.4867980771, −9.65964857368, −9.36042312795, −8.68880636198, −7.96799922500, −7.46340357329, −6.77181442827, −6.36067683503, −5.75753398469, −5.02219665130, −4.47608587083, −3.89251174955, −3.10294145796, −2.12408365654, 0,
2.12408365654, 3.10294145796, 3.89251174955, 4.47608587083, 5.02219665130, 5.75753398469, 6.36067683503, 6.77181442827, 7.46340357329, 7.96799922500, 8.68880636198, 9.36042312795, 9.65964857368, 10.4867980771, 10.8102860044, 11.1951872686, 11.8425963982, 12.5875643686, 12.6834405920, 12.8832229728, 13.6838905933, 14.0593119955, 14.5416374046, 14.9805811271, 15.6493611928
