| L(s) = 1 | + 2-s + 2·3-s − 4-s − 4·5-s + 2·6-s + 4·7-s − 3·8-s + 3·9-s − 4·10-s − 2·11-s − 2·12-s + 6·13-s + 4·14-s − 8·15-s − 16-s − 8·17-s + 3·18-s − 19-s + 4·20-s + 8·21-s − 2·22-s − 2·23-s − 6·24-s + 2·25-s + 6·26-s + 4·27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s + 1.51·7-s − 1.06·8-s + 9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s + 1.66·13-s + 1.06·14-s − 2.06·15-s − 1/4·16-s − 1.94·17-s + 0.707·18-s − 0.229·19-s + 0.894·20-s + 1.74·21-s − 0.426·22-s − 0.417·23-s − 1.22·24-s + 2/5·25-s + 1.17·26-s + 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.284280023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284280023\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−12.0798744008, −11.8739105144, −11.4085208449, −10.9289324032, −10.8415561337, −10.3317571499, −9.61000687688, −9.03211663271, −8.83746811854, −8.37436816344, −8.26351343286, −7.75989478086, −7.59950272568, −6.81360604684, −6.46787589567, −5.79402677262, −5.20430853708, −4.56991321081, −4.48660686111, −3.95264542154, −3.44365518363, −3.34310416861, −2.12198251427, −1.91788623711, −0.578917493177,
0.578917493177, 1.91788623711, 2.12198251427, 3.34310416861, 3.44365518363, 3.95264542154, 4.48660686111, 4.56991321081, 5.20430853708, 5.79402677262, 6.46787589567, 6.81360604684, 7.59950272568, 7.75989478086, 8.26351343286, 8.37436816344, 8.83746811854, 9.03211663271, 9.61000687688, 10.3317571499, 10.8415561337, 10.9289324032, 11.4085208449, 11.8739105144, 12.0798744008
