| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 5·7-s + 4·8-s + 11-s + 2·12-s − 10·13-s + 10·14-s + 8·16-s − 6·17-s − 7·19-s + 5·21-s + 2·22-s + 4·23-s + 4·24-s + 5·25-s − 20·26-s − 27-s + 10·28-s − 4·29-s + 7·31-s + 8·32-s + 33-s − 12·34-s − 7·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.88·7-s + 1.41·8-s + 0.301·11-s + 0.577·12-s − 2.77·13-s + 2.67·14-s + 2·16-s − 1.45·17-s − 1.60·19-s + 1.09·21-s + 0.426·22-s + 0.834·23-s + 0.816·24-s + 25-s − 3.92·26-s − 0.192·27-s + 1.88·28-s − 0.742·29-s + 1.25·31-s + 1.41·32-s + 0.174·33-s − 2.05·34-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.874352766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.874352766\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.76330192244983962234379002223, −11.99056174492223774297945854722, −11.64364226971689673781274977003, −11.12544473443726703312103083774, −10.57224368012032384238839533815, −10.35844163870448281900821259203, −9.502913131229102695737461291191, −8.966061030100122121943057064239, −8.183737458587139982514204139339, −8.142826525168905711157897447341, −7.23332851648736729301646181171, −7.04669903137814679268932128991, −6.35770667042513005377017322834, −5.19233923583365002529897989329, −4.97024898212468572981284269089, −4.58673912664646542633913166884, −4.26179664777318254873576045482, −3.16081551750051396315896111487, −2.21592468786932712240342476490, −1.89986761739282675830071919553,
1.89986761739282675830071919553, 2.21592468786932712240342476490, 3.16081551750051396315896111487, 4.26179664777318254873576045482, 4.58673912664646542633913166884, 4.97024898212468572981284269089, 5.19233923583365002529897989329, 6.35770667042513005377017322834, 7.04669903137814679268932128991, 7.23332851648736729301646181171, 8.142826525168905711157897447341, 8.183737458587139982514204139339, 8.966061030100122121943057064239, 9.502913131229102695737461291191, 10.35844163870448281900821259203, 10.57224368012032384238839533815, 11.12544473443726703312103083774, 11.64364226971689673781274977003, 11.99056174492223774297945854722, 12.76330192244983962234379002223
