| L(s) = 1 | + 3·3-s − 4·4-s + 13·7-s − 12·12-s − 13-s − 26·19-s + 39·21-s + 50·25-s − 27·27-s − 52·28-s − 26·31-s − 26·37-s − 3·39-s − 83·43-s + 49·49-s + 4·52-s − 78·57-s − 47·61-s + 64·64-s + 13·67-s + 286·73-s + 150·75-s + 104·76-s + 22·79-s − 81·81-s − 156·84-s − 13·91-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 13/7·7-s − 12-s − 0.0769·13-s − 1.36·19-s + 13/7·21-s + 2·25-s − 27-s − 1.85·28-s − 0.838·31-s − 0.702·37-s − 0.0769·39-s − 1.93·43-s + 49-s + 1/13·52-s − 1.36·57-s − 0.770·61-s + 64-s + 0.194·67-s + 3.91·73-s + 2·75-s + 1.36·76-s + 0.278·79-s − 81-s − 1.85·84-s − 1/7·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.301356741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301356741\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
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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
−16.44960962473725429953500678506, −15.33619547181759434000994251183, −14.79538314930110082820277115602, −14.68929085102026995003786521323, −13.96396072540954395904389671939, −13.67175235888688316267129445252, −12.84878159779038470756789436992, −12.33823488936215554299915870443, −11.20423937289260940719077991771, −11.05152271274972865795132837339, −10.08348895936540651189734201027, −9.171579227463954809412241897627, −8.547084336299271831896795172274, −8.434778742593843963541775978707, −7.60858119378173817575817595240, −6.57071045792209803133239001031, −5.07916973548642226261548216615, −4.76723639430079065409272750214, −3.60702769567781332238266215557, −2.04486508867934138690880670750,
2.04486508867934138690880670750, 3.60702769567781332238266215557, 4.76723639430079065409272750214, 5.07916973548642226261548216615, 6.57071045792209803133239001031, 7.60858119378173817575817595240, 8.434778742593843963541775978707, 8.547084336299271831896795172274, 9.171579227463954809412241897627, 10.08348895936540651189734201027, 11.05152271274972865795132837339, 11.20423937289260940719077991771, 12.33823488936215554299915870443, 12.84878159779038470756789436992, 13.67175235888688316267129445252, 13.96396072540954395904389671939, 14.68929085102026995003786521323, 14.79538314930110082820277115602, 15.33619547181759434000994251183, 16.44960962473725429953500678506
