| L(s) = 1 | − 7-s + 19-s + 25-s − 31-s + 2·37-s − 3·61-s + 3·73-s + 2·103-s + 109-s + 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 7-s + 19-s + 25-s − 31-s + 2·37-s − 3·61-s + 3·73-s + 2·103-s + 109-s + 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.168108148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168108148\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.033078272488603153747116717778, −9.020811866519435263649538136557, −8.321267328526487072880160028881, −7.77303004100725999372811788161, −7.66444432225361500905678299622, −7.34287298161908838889461920616, −6.60005625756178883466529574498, −6.59417055961223079887456440304, −6.11647421060675859243536855654, −5.73181894695275610502904723811, −5.24646355096021408915491488884, −4.87754967694813857178180932234, −4.43862406385607811620286990188, −3.95843649175983295343308045386, −3.30211812971448327960015430547, −3.25589680566919957042860187622, −2.65280056791359259628016203421, −2.13418449401187214349639258262, −1.39873304819395226299059943892, −0.71050136334443935450162667755,
0.71050136334443935450162667755, 1.39873304819395226299059943892, 2.13418449401187214349639258262, 2.65280056791359259628016203421, 3.25589680566919957042860187622, 3.30211812971448327960015430547, 3.95843649175983295343308045386, 4.43862406385607811620286990188, 4.87754967694813857178180932234, 5.24646355096021408915491488884, 5.73181894695275610502904723811, 6.11647421060675859243536855654, 6.59417055961223079887456440304, 6.60005625756178883466529574498, 7.34287298161908838889461920616, 7.66444432225361500905678299622, 7.77303004100725999372811788161, 8.321267328526487072880160028881, 9.020811866519435263649538136557, 9.033078272488603153747116717778
