| L(s) = 1 | − 4·5-s + 6·9-s + 6·13-s + 2·17-s + 14·25-s + 10·29-s + 2·37-s + 10·41-s − 24·45-s + 14·49-s + 28·53-s + 10·61-s − 24·65-s + 12·73-s + 9·81-s − 8·85-s − 20·89-s − 36·97-s + 2·101-s − 24·109-s − 14·113-s + 36·117-s + 22·121-s − 64·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2·9-s + 1.66·13-s + 0.485·17-s + 14/5·25-s + 1.85·29-s + 0.328·37-s + 1.56·41-s − 3.57·45-s + 2·49-s + 3.84·53-s + 1.28·61-s − 2.97·65-s + 1.40·73-s + 81-s − 0.867·85-s − 2.11·89-s − 3.65·97-s + 0.199·101-s − 2.29·109-s − 1.31·113-s + 3.32·117-s + 2·121-s − 5.72·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.32456289254254282086171325256, −7.11478437130649698482413304070, −6.87360197021863132651143148562, −6.79139461715931586476520847496, −6.67124480121587678737305448806, −6.25588444438717162541363592433, −5.79282805802190037060522883054, −5.77039539865653728489539145373, −5.55435006928759479084408583914, −5.12764861783047012396148875616, −4.93694287971795001527906351117, −4.52353758887044459762626346130, −4.43843951632667894305509682956, −4.03567161531092997170504951083, −3.87615301317608835419108432422, −3.85499276898535288409211173056, −3.71032316470829712170155206853, −3.16518765702227992097418965696, −2.62214148073303372759935715417, −2.51782263686722652720846050597, −2.48251574095226401889594628541, −1.44294101272403318532430877000, −1.18668516801452826381590313233, −1.10320839767098872391392769323, −0.60562780862176733974533240617,
0.60562780862176733974533240617, 1.10320839767098872391392769323, 1.18668516801452826381590313233, 1.44294101272403318532430877000, 2.48251574095226401889594628541, 2.51782263686722652720846050597, 2.62214148073303372759935715417, 3.16518765702227992097418965696, 3.71032316470829712170155206853, 3.85499276898535288409211173056, 3.87615301317608835419108432422, 4.03567161531092997170504951083, 4.43843951632667894305509682956, 4.52353758887044459762626346130, 4.93694287971795001527906351117, 5.12764861783047012396148875616, 5.55435006928759479084408583914, 5.77039539865653728489539145373, 5.79282805802190037060522883054, 6.25588444438717162541363592433, 6.67124480121587678737305448806, 6.79139461715931586476520847496, 6.87360197021863132651143148562, 7.11478437130649698482413304070, 7.32456289254254282086171325256
