| L(s) = 1 | + 2·3-s + 3·9-s − 6·13-s + 8·17-s − 12·23-s − 25-s + 4·27-s + 8·29-s − 12·39-s + 24·43-s + 14·49-s + 16·51-s − 4·53-s − 20·61-s − 24·69-s − 2·75-s + 16·79-s + 5·81-s + 16·87-s + 16·101-s − 8·103-s + 24·113-s − 18·117-s + 6·121-s + 127-s + 48·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.66·13-s + 1.94·17-s − 2.50·23-s − 1/5·25-s + 0.769·27-s + 1.48·29-s − 1.92·39-s + 3.65·43-s + 2·49-s + 2.24·51-s − 0.549·53-s − 2.56·61-s − 2.88·69-s − 0.230·75-s + 1.80·79-s + 5/9·81-s + 1.71·87-s + 1.59·101-s − 0.788·103-s + 2.25·113-s − 1.66·117-s + 6/11·121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.976721051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.976721051\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.802489517995736578895768685022, −8.591944056453795253319972646796, −7.85124402113655299673266418240, −7.71346959763769979924482794687, −7.65616445214772453499077625994, −7.33506465559475082985252835237, −6.65251308736222036694854106652, −6.22747184174198455283150562283, −5.70844355813211803236576906409, −5.67471881743627967381059555314, −4.88007608825958605642162839299, −4.52714613205209160877368363729, −4.15565375873922002938807711457, −3.77342556323853557889501250049, −3.17799836438729921311517688163, −2.84637015589350344038858008943, −2.21986757086165192794457423524, −2.14878127532476800078903135475, −1.22751300856739268010341789854, −0.60201821950781650718441487946,
0.60201821950781650718441487946, 1.22751300856739268010341789854, 2.14878127532476800078903135475, 2.21986757086165192794457423524, 2.84637015589350344038858008943, 3.17799836438729921311517688163, 3.77342556323853557889501250049, 4.15565375873922002938807711457, 4.52714613205209160877368363729, 4.88007608825958605642162839299, 5.67471881743627967381059555314, 5.70844355813211803236576906409, 6.22747184174198455283150562283, 6.65251308736222036694854106652, 7.33506465559475082985252835237, 7.65616445214772453499077625994, 7.71346959763769979924482794687, 7.85124402113655299673266418240, 8.591944056453795253319972646796, 8.802489517995736578895768685022
