| L(s) = 1 | + 2-s + 2·4-s + 2·5-s − 2·7-s + 5·8-s + 2·10-s − 2·11-s − 7·13-s − 2·14-s + 5·16-s − 7·17-s + 6·19-s + 4·20-s − 2·22-s − 6·23-s − 7·25-s − 7·26-s − 4·28-s − 29-s + 8·31-s + 10·32-s − 7·34-s − 4·35-s − 37-s + 6·38-s + 10·40-s + 9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.894·5-s − 0.755·7-s + 1.76·8-s + 0.632·10-s − 0.603·11-s − 1.94·13-s − 0.534·14-s + 5/4·16-s − 1.69·17-s + 1.37·19-s + 0.894·20-s − 0.426·22-s − 1.25·23-s − 7/5·25-s − 1.37·26-s − 0.755·28-s − 0.185·29-s + 1.43·31-s + 1.76·32-s − 1.20·34-s − 0.676·35-s − 0.164·37-s + 0.973·38-s + 1.58·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797136180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797136180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.71583373732554622146040930695, −13.48498487929867655058308897439, −12.87183096216739190185826885775, −12.32825400348559817062130598804, −11.61694792247288167269886733238, −11.55009927918403594947634959404, −10.42252617046725000405145553302, −10.28989878879188507205957631370, −9.623652786259858771580091256995, −9.435674583513402062223853553021, −8.080135195646394940869329121953, −7.77585513134230575802964807258, −7.00209494575348991887569003457, −6.63571917544642377846356517840, −5.88321458374189873299594450808, −5.20143778442319878796371214143, −4.63689197056435120495687654562, −3.77137715802230625281806667228, −2.36681532156722348274801924146, −2.28342814195701605346487158022,
2.28342814195701605346487158022, 2.36681532156722348274801924146, 3.77137715802230625281806667228, 4.63689197056435120495687654562, 5.20143778442319878796371214143, 5.88321458374189873299594450808, 6.63571917544642377846356517840, 7.00209494575348991887569003457, 7.77585513134230575802964807258, 8.080135195646394940869329121953, 9.435674583513402062223853553021, 9.623652786259858771580091256995, 10.28989878879188507205957631370, 10.42252617046725000405145553302, 11.55009927918403594947634959404, 11.61694792247288167269886733238, 12.32825400348559817062130598804, 12.87183096216739190185826885775, 13.48498487929867655058308897439, 13.71583373732554622146040930695
