L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s + 2·12-s + 16-s + 12·17-s − 3·18-s − 8·19-s − 2·24-s + 25-s + 4·27-s − 32-s − 12·34-s + 3·36-s + 8·38-s − 12·41-s − 8·43-s + 2·48-s + 2·49-s − 50-s + 24·51-s − 4·54-s − 16·57-s + 64-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 9-s + 0.577·12-s + 1/4·16-s + 2.91·17-s − 0.707·18-s − 1.83·19-s − 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.176·32-s − 2.05·34-s + 1/2·36-s + 1.29·38-s − 1.87·41-s − 1.21·43-s + 0.288·48-s + 2/7·49-s − 0.141·50-s + 3.36·51-s − 0.544·54-s − 2.11·57-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.372802002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.372802002\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39160509435207624542212960871, −10.10989512091566913391448478748, −9.305587122869086271308905422277, −9.082506062594783472831133921500, −8.307525687782695431948900879973, −7.941231322257043910223245152113, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −6.42217617652666865799421983967, −5.48025531888955534300415720953, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −3.36585804145949552210873743484, −2.39041205999551187286748903960, −1.43294542786364492056174526795,
1.43294542786364492056174526795, 2.39041205999551187286748903960, 3.36585804145949552210873743484, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 5.48025531888955534300415720953, 6.42217617652666865799421983967, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 7.941231322257043910223245152113, 8.307525687782695431948900879973, 9.082506062594783472831133921500, 9.305587122869086271308905422277, 10.10989512091566913391448478748, 10.39160509435207624542212960871