L(s) = 1 | − 3-s + 4·5-s + 9-s − 4·15-s + 8·19-s + 2·25-s − 27-s + 4·29-s − 8·43-s + 4·45-s − 16·47-s + 2·49-s + 20·53-s − 8·57-s − 8·67-s + 32·71-s − 12·73-s − 2·75-s + 81-s − 4·87-s + 32·95-s − 28·97-s − 12·101-s − 6·121-s − 28·125-s + 127-s + 8·129-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.03·15-s + 1.83·19-s + 2/5·25-s − 0.192·27-s + 0.742·29-s − 1.21·43-s + 0.596·45-s − 2.33·47-s + 2/7·49-s + 2.74·53-s − 1.05·57-s − 0.977·67-s + 3.79·71-s − 1.40·73-s − 0.230·75-s + 1/9·81-s − 0.428·87-s + 3.28·95-s − 2.84·97-s − 1.19·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.636513767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636513767\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.826637660192491869357018654726, −9.629712905768100889040064075295, −9.406971771311537534006960334033, −8.391046533691483698004382219288, −8.131897207027853144179789700526, −7.17212008872630953090710242182, −6.83791452023955841983204630481, −6.23593377615022317750995350027, −5.56983356275681900785379754785, −5.42460705368026111370695370006, −4.77602747706575131165960838209, −3.81654902962345598726342363363, −2.97479066673588337231673992698, −2.09803110754033007053716721218, −1.26883614185569776706713350485,
1.26883614185569776706713350485, 2.09803110754033007053716721218, 2.97479066673588337231673992698, 3.81654902962345598726342363363, 4.77602747706575131165960838209, 5.42460705368026111370695370006, 5.56983356275681900785379754785, 6.23593377615022317750995350027, 6.83791452023955841983204630481, 7.17212008872630953090710242182, 8.131897207027853144179789700526, 8.391046533691483698004382219288, 9.406971771311537534006960334033, 9.629712905768100889040064075295, 9.826637660192491869357018654726