| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 3·9-s + 2·12-s + 4·14-s + 16-s − 3·18-s − 8·19-s − 8·21-s − 2·24-s + 25-s + 4·27-s − 4·28-s − 12·29-s + 16·31-s − 32-s + 3·36-s + 4·37-s + 8·38-s + 8·42-s + 2·48-s + 9·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 9-s + 0.577·12-s + 1.06·14-s + 1/4·16-s − 0.707·18-s − 1.83·19-s − 1.74·21-s − 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.755·28-s − 2.22·29-s + 2.87·31-s − 0.176·32-s + 1/2·36-s + 0.657·37-s + 1.29·38-s + 1.23·42-s + 0.288·48-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.465521936756471995013087083461, −7.963508266137095813927181717979, −7.941231322257043910223245152113, −7.13186260729171705377151627447, −6.59025842620406930577287885090, −6.42217617652666865799421983967, −5.96167843576973109324683088583, −5.08035454069229832684176549713, −4.36922820730120103870866884845, −3.84605851270934769325068681916, −3.36585804145949552210873743484, −2.58646024824692464345493568105, −2.40188042712462214345165355298, −1.34432245463299551756265640292, 0,
1.34432245463299551756265640292, 2.40188042712462214345165355298, 2.58646024824692464345493568105, 3.36585804145949552210873743484, 3.84605851270934769325068681916, 4.36922820730120103870866884845, 5.08035454069229832684176549713, 5.96167843576973109324683088583, 6.42217617652666865799421983967, 6.59025842620406930577287885090, 7.13186260729171705377151627447, 7.941231322257043910223245152113, 7.963508266137095813927181717979, 8.465521936756471995013087083461
