| L(s) = 1 | + 4-s − 5-s + 5·7-s + 9-s + 11-s − 5·17-s − 19-s − 20-s + 5·28-s − 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s + 5·63-s − 5·68-s − 76-s + 5·77-s + 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s + 5·7-s + 9-s + 11-s − 5·17-s − 19-s − 20-s + 5·28-s − 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s + 5·63-s − 5·68-s − 76-s + 5·77-s + 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804113662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804113662\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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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
−7.25824870409824046629302395546, −7.25295890388695595125625171485, −6.84197893935016528364593251708, −6.81245314741722414884067953495, −6.62799677270334789006967904469, −6.05309360147585951832195296621, −6.03483358937890099689493189535, −6.02246635816760699225180128623, −5.21953703049033772856713006860, −5.12876743556867126262837623481, −4.88277666701319250064485140216, −4.67497466134839374078109652980, −4.50273962629193106970797731922, −4.38912833790693122430251876095, −4.22382880013617333151397361241, −4.01018109915690361774245627949, −3.83684259835961469185248358774, −3.27676253265898877875472523749, −2.52399645412985078657528973218, −2.52164043880825133351640873521, −2.17532612982759790135681592626, −1.87194181839078801009391574155, −1.79974125629603895484407567238, −1.53864732888383593462439974632, −1.10269901584240731066200010138,
1.10269901584240731066200010138, 1.53864732888383593462439974632, 1.79974125629603895484407567238, 1.87194181839078801009391574155, 2.17532612982759790135681592626, 2.52164043880825133351640873521, 2.52399645412985078657528973218, 3.27676253265898877875472523749, 3.83684259835961469185248358774, 4.01018109915690361774245627949, 4.22382880013617333151397361241, 4.38912833790693122430251876095, 4.50273962629193106970797731922, 4.67497466134839374078109652980, 4.88277666701319250064485140216, 5.12876743556867126262837623481, 5.21953703049033772856713006860, 6.02246635816760699225180128623, 6.03483358937890099689493189535, 6.05309360147585951832195296621, 6.62799677270334789006967904469, 6.81245314741722414884067953495, 6.84197893935016528364593251708, 7.25295890388695595125625171485, 7.25824870409824046629302395546
