| L(s) = 1 | + 3-s + 4·5-s − 5·7-s − 6·11-s + 10·13-s + 4·15-s − 2·17-s − 19-s − 5·21-s + 6·23-s + 5·25-s − 27-s + 3·31-s − 6·33-s − 20·35-s − 3·37-s + 10·39-s − 12·41-s + 10·43-s + 4·47-s + 18·49-s − 2·51-s + 6·53-s − 24·55-s − 57-s + 6·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1.88·7-s − 1.80·11-s + 2.77·13-s + 1.03·15-s − 0.485·17-s − 0.229·19-s − 1.09·21-s + 1.25·23-s + 25-s − 0.192·27-s + 0.538·31-s − 1.04·33-s − 3.38·35-s − 0.493·37-s + 1.60·39-s − 1.87·41-s + 1.52·43-s + 0.583·47-s + 18/7·49-s − 0.280·51-s + 0.824·53-s − 3.23·55-s − 0.132·57-s + 0.781·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.563481723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.563481723\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.53439285514371712800488479945, −10.26154474355818601676936063414, −9.918930137419883164237072481204, −9.282217740634440571607027521347, −9.157532573971884410867581024678, −8.661694721472449773927791108720, −8.315504782802067696491074355328, −7.77660233012000758789839311064, −6.98455569535303703322178211421, −6.47595108587198978840124893890, −6.45315272525040940957781268607, −5.71231980134881970389951878398, −5.53932583553114272664537723585, −5.01101152974708216594616006829, −3.80764450688253338559107168897, −3.70796240581125254299915098261, −2.85037097479529076476818632112, −2.59401460203519877823684274842, −1.88409642934232701845675934704, −0.835594766374436951853173964543,
0.835594766374436951853173964543, 1.88409642934232701845675934704, 2.59401460203519877823684274842, 2.85037097479529076476818632112, 3.70796240581125254299915098261, 3.80764450688253338559107168897, 5.01101152974708216594616006829, 5.53932583553114272664537723585, 5.71231980134881970389951878398, 6.45315272525040940957781268607, 6.47595108587198978840124893890, 6.98455569535303703322178211421, 7.77660233012000758789839311064, 8.315504782802067696491074355328, 8.661694721472449773927791108720, 9.157532573971884410867581024678, 9.282217740634440571607027521347, 9.918930137419883164237072481204, 10.26154474355818601676936063414, 10.53439285514371712800488479945
