| L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 2·7-s − 4·8-s + 2·10-s + 5·11-s + 13-s + 4·14-s + 5·16-s − 3·20-s − 10·22-s + 23-s + 2·25-s − 2·26-s − 6·28-s + 3·29-s + 17·31-s − 6·32-s + 2·35-s − 2·37-s + 4·40-s − 17·41-s − 6·43-s + 15·44-s − 2·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.447·5-s − 0.755·7-s − 1.41·8-s + 0.632·10-s + 1.50·11-s + 0.277·13-s + 1.06·14-s + 5/4·16-s − 0.670·20-s − 2.13·22-s + 0.208·23-s + 2/5·25-s − 0.392·26-s − 1.13·28-s + 0.557·29-s + 3.05·31-s − 1.06·32-s + 0.338·35-s − 0.328·37-s + 0.632·40-s − 2.65·41-s − 0.914·43-s + 2.26·44-s − 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8743695126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8743695126\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 3 T - T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T + 133 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 17 T + 153 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 211 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.46362769212668555174233177355, −10.16084379052051402647932147893, −9.879949479711905537275271281386, −9.535346551387154414694290683943, −8.867376536083673390760119910492, −8.638109233435591597672701580597, −8.234962024644255586792061168563, −7.956053490775710793649126002140, −7.14044343528659801100496828380, −6.66905150130503432360910126425, −6.43945258485775047281224833849, −6.41681475910582889607095725592, −5.21403278492601116080479391320, −4.94521540255671770583730319194, −3.89633259338865014135668912905, −3.65445389378952927062629006311, −2.96807548835528318907132220424, −2.29679447527135897016348658064, −1.35434847827472147039815903809, −0.70293052423832835508495182842,
0.70293052423832835508495182842, 1.35434847827472147039815903809, 2.29679447527135897016348658064, 2.96807548835528318907132220424, 3.65445389378952927062629006311, 3.89633259338865014135668912905, 4.94521540255671770583730319194, 5.21403278492601116080479391320, 6.41681475910582889607095725592, 6.43945258485775047281224833849, 6.66905150130503432360910126425, 7.14044343528659801100496828380, 7.956053490775710793649126002140, 8.234962024644255586792061168563, 8.638109233435591597672701580597, 8.867376536083673390760119910492, 9.535346551387154414694290683943, 9.879949479711905537275271281386, 10.16084379052051402647932147893, 10.46362769212668555174233177355
