| L(s) = 1 | − 4-s + 4·7-s + 16-s − 25-s − 4·28-s + 4·31-s + 10·49-s − 64-s − 4·73-s + 100-s + 4·112-s − 121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s + 4·7-s + 16-s − 25-s − 4·28-s + 4·31-s + 10·49-s − 64-s − 4·73-s + 100-s + 4·112-s − 121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.266038298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266038298\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.580414721650511879668141091532, −8.332597739475181929107133254285, −8.164318686580281447868451982385, −7.961816500586970972920250543258, −7.43773208708229555394699808320, −7.36518257735511160934061330984, −6.62835101086678887862020391294, −5.93729789456770226597164268036, −5.87011552526503377948278442886, −5.17147362267924256959092263516, −5.09413490146369778547891403753, −4.50394828099472526858382166202, −4.48338386408034951493369863720, −4.21384490743126231566117374288, −3.61688609705468385581572613931, −2.67413679028139330027831027805, −2.50981962471622191764769163691, −1.76413640980303348561429978669, −1.19201641050947408877953704260, −1.15558673807892511001808746899,
1.15558673807892511001808746899, 1.19201641050947408877953704260, 1.76413640980303348561429978669, 2.50981962471622191764769163691, 2.67413679028139330027831027805, 3.61688609705468385581572613931, 4.21384490743126231566117374288, 4.48338386408034951493369863720, 4.50394828099472526858382166202, 5.09413490146369778547891403753, 5.17147362267924256959092263516, 5.87011552526503377948278442886, 5.93729789456770226597164268036, 6.62835101086678887862020391294, 7.36518257735511160934061330984, 7.43773208708229555394699808320, 7.961816500586970972920250543258, 8.164318686580281447868451982385, 8.332597739475181929107133254285, 8.580414721650511879668141091532
