| L(s) = 1 | − 2·3-s + 4-s − 4·7-s + 9-s − 2·12-s + 6·13-s + 16-s + 4·19-s + 8·21-s − 25-s + 4·27-s − 4·28-s + 8·31-s + 36-s − 12·39-s − 8·43-s − 2·48-s − 49-s + 6·52-s − 8·57-s − 8·61-s − 4·63-s + 64-s − 8·67-s − 10·73-s + 2·75-s + 4·76-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.917·19-s + 1.74·21-s − 1/5·25-s + 0.769·27-s − 0.755·28-s + 1.43·31-s + 1/6·36-s − 1.92·39-s − 1.21·43-s − 0.288·48-s − 1/7·49-s + 0.832·52-s − 1.05·57-s − 1.02·61-s − 0.503·63-s + 1/8·64-s − 0.977·67-s − 1.17·73-s + 0.230·75-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−7.74151171551449502330125125699, −7.16152902049794544132675437202, −6.70040122798654550620733112587, −6.40818821278537747508017147952, −6.10065510704317972095508908800, −5.81156544978872936451623421426, −5.26638380188506621374707060834, −4.71491570384763710121902296680, −4.16198853833206437749890734961, −3.42665838600910385696912032166, −3.19110399319157223137105548532, −2.68985241690590522148729296526, −1.59651631208303436278229670138, −1.00402404230437196821566182156, 0,
1.00402404230437196821566182156, 1.59651631208303436278229670138, 2.68985241690590522148729296526, 3.19110399319157223137105548532, 3.42665838600910385696912032166, 4.16198853833206437749890734961, 4.71491570384763710121902296680, 5.26638380188506621374707060834, 5.81156544978872936451623421426, 6.10065510704317972095508908800, 6.40818821278537747508017147952, 6.70040122798654550620733112587, 7.16152902049794544132675437202, 7.74151171551449502330125125699
