| L(s) = 1 | − 6·11-s − 4·13-s − 4·19-s − 6·23-s + 5·25-s − 12·29-s + 8·31-s − 2·37-s + 24·41-s − 8·43-s − 12·47-s − 6·53-s − 10·61-s − 8·67-s − 12·71-s − 10·73-s + 4·79-s − 24·83-s − 12·89-s + 20·97-s + 12·101-s + 8·103-s − 6·107-s − 14·109-s + 12·113-s + 11·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.80·11-s − 1.10·13-s − 0.917·19-s − 1.25·23-s + 25-s − 2.22·29-s + 1.43·31-s − 0.328·37-s + 3.74·41-s − 1.21·43-s − 1.75·47-s − 0.824·53-s − 1.28·61-s − 0.977·67-s − 1.42·71-s − 1.17·73-s + 0.450·79-s − 2.63·83-s − 1.27·89-s + 2.03·97-s + 1.19·101-s + 0.788·103-s − 0.580·107-s − 1.34·109-s + 1.12·113-s + 121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2284518589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2284518589\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.554574195003549911579618090288, −9.057669002120242481862504988676, −8.773650660616631093854656284318, −8.115740235985354181582674125923, −7.960665559929633233705723364521, −7.43459518678917806391853825802, −7.42644860065159425037676445151, −6.72575848984387405197821086068, −6.20080554217265571732577823089, −5.74946873491688202489108732054, −5.64655303408395581646025433378, −4.76355785055075911897485270082, −4.69279063319515618283618631482, −4.27818700991027018142613131213, −3.55129892550125268171494229195, −2.85389113381592196300907702039, −2.68751287498481909050364269498, −2.07900445752771484411713967524, −1.46082212216262558740401482019, −0.17057248429115436909393366527,
0.17057248429115436909393366527, 1.46082212216262558740401482019, 2.07900445752771484411713967524, 2.68751287498481909050364269498, 2.85389113381592196300907702039, 3.55129892550125268171494229195, 4.27818700991027018142613131213, 4.69279063319515618283618631482, 4.76355785055075911897485270082, 5.64655303408395581646025433378, 5.74946873491688202489108732054, 6.20080554217265571732577823089, 6.72575848984387405197821086068, 7.42644860065159425037676445151, 7.43459518678917806391853825802, 7.960665559929633233705723364521, 8.115740235985354181582674125923, 8.773650660616631093854656284318, 9.057669002120242481862504988676, 9.554574195003549911579618090288
