| L(s) = 1 | − 2·3-s + 9-s − 4·19-s + 4·27-s − 20·43-s + 14·49-s + 8·57-s + 28·67-s − 4·73-s − 11·81-s + 20·97-s + 14·121-s + 127-s + 40·129-s + 131-s + 137-s + 139-s − 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 4·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.917·19-s + 0.769·27-s − 3.04·43-s + 2·49-s + 1.05·57-s + 3.42·67-s − 0.468·73-s − 1.22·81-s + 2.03·97-s + 1.27·121-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.30·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 0.305·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036133094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036133094\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.107897721582570851156740561348, −8.702997871714932983765547736546, −8.298527165408829152969619472751, −8.228370084764894638761475578086, −7.50202561122533516200348335277, −7.13742115109991521178988712770, −6.64004200905974742111765148397, −6.54549270087613272619010481884, −6.08244806334364017350746382929, −5.53048650246419820712533709257, −5.29915590012214749850616505894, −4.95976962626720630007075593800, −4.28684841584362602698475383144, −4.17880777761139982658805710114, −3.26141843624389429733080103779, −3.20878852499132098605088769285, −2.09261026176567940835496866639, −2.07390533808297228106569892821, −1.02709656310998515454024541371, −0.43629136808868055354532618154,
0.43629136808868055354532618154, 1.02709656310998515454024541371, 2.07390533808297228106569892821, 2.09261026176567940835496866639, 3.20878852499132098605088769285, 3.26141843624389429733080103779, 4.17880777761139982658805710114, 4.28684841584362602698475383144, 4.95976962626720630007075593800, 5.29915590012214749850616505894, 5.53048650246419820712533709257, 6.08244806334364017350746382929, 6.54549270087613272619010481884, 6.64004200905974742111765148397, 7.13742115109991521178988712770, 7.50202561122533516200348335277, 8.228370084764894638761475578086, 8.298527165408829152969619472751, 8.702997871714932983765547736546, 9.107897721582570851156740561348
