L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s + 4·8-s + 2·9-s + 4·10-s + 2·11-s − 6·12-s − 2·13-s + 4·14-s − 4·15-s + 5·16-s − 4·17-s + 4·18-s − 10·19-s + 6·20-s − 4·21-s + 4·22-s + 8·23-s − 8·24-s − 2·25-s − 4·26-s − 6·27-s + 6·28-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.970·17-s + 0.942·18-s − 2.29·19-s + 1.34·20-s − 0.872·21-s + 0.852·22-s + 1.66·23-s − 1.63·24-s − 2/5·25-s − 0.784·26-s − 1.15·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.248806007\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.248806007\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−13.04142939108390237862827135983, −13.02165373672923789671526488169, −12.02573754642256028029402201417, −11.95456775219894293344517668118, −11.24696787726035266900578757471, −10.97803505558155713549145959874, −10.46813122479265160040897078207, −9.971163456632217281631794963600, −9.141839153069202927093459272189, −8.595221279802153733551270833364, −7.80396324611489422793705431355, −6.94310403447038207708400147483, −6.52527085313281038885199593615, −6.23746833597972884543119043828, −5.40100649640324540624234594736, −5.03379623435561269802443819410, −4.45294812940400001555688902048, −3.80599127239261113785174017657, −2.44971458371973372755039334054, −1.77756563632127092242172589873,
1.77756563632127092242172589873, 2.44971458371973372755039334054, 3.80599127239261113785174017657, 4.45294812940400001555688902048, 5.03379623435561269802443819410, 5.40100649640324540624234594736, 6.23746833597972884543119043828, 6.52527085313281038885199593615, 6.94310403447038207708400147483, 7.80396324611489422793705431355, 8.595221279802153733551270833364, 9.141839153069202927093459272189, 9.971163456632217281631794963600, 10.46813122479265160040897078207, 10.97803505558155713549145959874, 11.24696787726035266900578757471, 11.95456775219894293344517668118, 12.02573754642256028029402201417, 13.02165373672923789671526488169, 13.04142939108390237862827135983