| L(s) = 1 | + 3·4-s − 4·7-s − 2·13-s + 5·16-s − 19-s − 8·25-s − 12·28-s + 8·31-s + 6·37-s + 2·49-s − 6·52-s − 16·61-s + 3·64-s − 20·67-s + 4·73-s − 3·76-s + 4·79-s + 8·91-s − 18·97-s − 24·100-s + 12·103-s + 22·109-s − 20·112-s + 4·121-s + 24·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.51·7-s − 0.554·13-s + 5/4·16-s − 0.229·19-s − 8/5·25-s − 2.26·28-s + 1.43·31-s + 0.986·37-s + 2/7·49-s − 0.832·52-s − 2.04·61-s + 3/8·64-s − 2.44·67-s + 0.468·73-s − 0.344·76-s + 0.450·79-s + 0.838·91-s − 1.82·97-s − 2.39·100-s + 1.18·103-s + 2.10·109-s − 1.88·112-s + 4/11·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.107395110056093026434156650477, −7.55244798122753458130865425531, −7.37329995126136806994410174998, −6.83910512728892399521935566081, −6.28548306673646853527100381142, −6.02950278814659044351983412645, −5.91349134597759617323525459700, −4.86517006327307671822610956547, −4.44541312687435534988529591311, −3.65666760533142611983188590748, −3.17521013186538179753526943634, −2.67612331000190134273821497347, −2.21398502546222424039556682543, −1.36419561150181600465757028761, 0,
1.36419561150181600465757028761, 2.21398502546222424039556682543, 2.67612331000190134273821497347, 3.17521013186538179753526943634, 3.65666760533142611983188590748, 4.44541312687435534988529591311, 4.86517006327307671822610956547, 5.91349134597759617323525459700, 6.02950278814659044351983412645, 6.28548306673646853527100381142, 6.83910512728892399521935566081, 7.37329995126136806994410174998, 7.55244798122753458130865425531, 8.107395110056093026434156650477
