L(s) = 1 | + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s − 2·29-s − 2·41-s − 2·45-s + 4·49-s + 2·53-s + 2·61-s − 4·65-s + 2·73-s + 81-s − 4·85-s − 2·89-s − 2·97-s − 2·101-s − 2·109-s + 2·117-s + 4·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
L(s) = 1 | + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s − 2·29-s − 2·41-s − 2·45-s + 4·49-s + 2·53-s + 2·61-s − 4·65-s + 2·73-s + 81-s − 4·85-s − 2·89-s − 2·97-s − 2·101-s − 2·109-s + 2·117-s + 4·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7031794759\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7031794759\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−7.968856433519597690605333638530, −7.60040728919371533297932533832, −7.17375748317411581951368264135, −7.15947266291594548559820984459, −6.91479889423352608118592797952, −6.74703446302938217338749664756, −6.65820829769792790057472118000, −6.15609306423635503235058732235, −5.96510884259150119521028902577, −5.49755724781655768682368337989, −5.44132731449757688771682563388, −5.32969246559001744155013287283, −5.30889111847725754477391362922, −4.84145422907222959210007397475, −4.27624416230100452575007135126, −4.16908640299428815677374098549, −4.14650064633458554788531923730, −3.46141081493330175932817934696, −3.18827094450451246200727851099, −2.68355143182558775553460465950, −2.54060602965108863061103159857, −2.18537959018974700229037322651, −2.12173342786610617706684343655, −1.78396047555416462650800385223, −0.921356769282575708589389190251,
0.921356769282575708589389190251, 1.78396047555416462650800385223, 2.12173342786610617706684343655, 2.18537959018974700229037322651, 2.54060602965108863061103159857, 2.68355143182558775553460465950, 3.18827094450451246200727851099, 3.46141081493330175932817934696, 4.14650064633458554788531923730, 4.16908640299428815677374098549, 4.27624416230100452575007135126, 4.84145422907222959210007397475, 5.30889111847725754477391362922, 5.32969246559001744155013287283, 5.44132731449757688771682563388, 5.49755724781655768682368337989, 5.96510884259150119521028902577, 6.15609306423635503235058732235, 6.65820829769792790057472118000, 6.74703446302938217338749664756, 6.91479889423352608118592797952, 7.15947266291594548559820984459, 7.17375748317411581951368264135, 7.60040728919371533297932533832, 7.968856433519597690605333638530