| L(s) = 1 | + 4-s − 2·7-s + 2·13-s − 2·19-s + 4·25-s − 2·28-s + 49-s + 2·52-s + 2·61-s − 2·76-s + 2·79-s − 4·91-s + 2·97-s + 4·100-s − 4·103-s + 4·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | + 4-s − 2·7-s + 2·13-s − 2·19-s + 4·25-s − 2·28-s + 49-s + 2·52-s + 2·61-s − 2·76-s + 2·79-s − 4·91-s + 2·97-s + 4·100-s − 4·103-s + 4·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9628087041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9628087041\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 103 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
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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
−7.18551444495452227673406772615, −7.09027976863346583663906770029, −6.90755395880477267551042055507, −6.65956504988089730504825770078, −6.43681024640810104622541348410, −6.43069731405788428049546904375, −6.38157293621589072006005990935, −5.89831921974029083967596286995, −5.84419107674818716313263069884, −5.44016798597709154893533782543, −4.98298982328206976453977122362, −4.92433245517473811817350748300, −4.82729322277967008162410716621, −4.15776465182907152916558322417, −3.98788792494418015169251516367, −3.95424233861339061011439569191, −3.35135815146689238618968971753, −3.32449946038957578896279548384, −3.13738861457169047996351408440, −2.65158796232566032746354303934, −2.44799986758828544214896780038, −2.27500541487441074042644019185, −1.74545169287564529552992587700, −1.17512664990643477365639614746, −0.939426999057368386969181400032,
0.939426999057368386969181400032, 1.17512664990643477365639614746, 1.74545169287564529552992587700, 2.27500541487441074042644019185, 2.44799986758828544214896780038, 2.65158796232566032746354303934, 3.13738861457169047996351408440, 3.32449946038957578896279548384, 3.35135815146689238618968971753, 3.95424233861339061011439569191, 3.98788792494418015169251516367, 4.15776465182907152916558322417, 4.82729322277967008162410716621, 4.92433245517473811817350748300, 4.98298982328206976453977122362, 5.44016798597709154893533782543, 5.84419107674818716313263069884, 5.89831921974029083967596286995, 6.38157293621589072006005990935, 6.43069731405788428049546904375, 6.43681024640810104622541348410, 6.65956504988089730504825770078, 6.90755395880477267551042055507, 7.09027976863346583663906770029, 7.18551444495452227673406772615
