| L(s) = 1 | − 26·7-s + 60·13-s − 26·19-s − 48·25-s + 6·31-s − 34·37-s + 340·43-s + 409·49-s + 144·61-s + 86·67-s + 190·73-s + 138·79-s − 1.56e3·91-s + 64·97-s − 122·103-s + 130·109-s − 192·121-s + 127-s + 131-s + 676·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.71·7-s + 4.61·13-s − 1.36·19-s − 1.91·25-s + 6/31·31-s − 0.918·37-s + 7.90·43-s + 8.34·49-s + 2.36·61-s + 1.28·67-s + 2.60·73-s + 1.74·79-s − 17.1·91-s + 0.659·97-s − 1.18·103-s + 1.19·109-s − 1.58·121-s + 0.00787·127-s + 0.00763·131-s + 5.08·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.120566779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120566779\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 48 T^{2} + 1679 T^{4} + 48 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 192 T^{2} + 22223 T^{4} + 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 450 T^{2} + 118979 T^{4} + 450 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 13 T - 192 T^{2} + 13 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 546 T^{2} + 18275 T^{4} + 546 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1170 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 3 T - 952 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 17 T - 1080 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3136 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 85 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 784 T^{2} - 4265025 T^{4} - 784 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4466 T^{2} + 12054675 T^{4} + 4466 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 1230 T^{2} - 10604461 T^{4} - 1230 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 72 T + 1463 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 43 T - 2640 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7344 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 95 T + 3696 T^{2} - 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 69 T - 1480 T^{2} - 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10080 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.90669347494207045335594061458, −6.48798977425343006227048134791, −6.40016633293207713285343821173, −6.34545777139186181357778014479, −5.99940538329425796228326822905, −5.83897559115442374706069078168, −5.70296552696148730013076207314, −5.61388162320183071262403076263, −5.56328869754516015083940980932, −4.57042721074303415721021521765, −4.49937841827387046102345269952, −4.06002108962734619602669889331, −4.03489177231261257232370131593, −3.68947910265967532936563385404, −3.53341609748013895997648511450, −3.51724645317691727385229881191, −3.35692932381284964919021160274, −2.63184626224489193992625927549, −2.49134454880760851150475675657, −2.15029142113237372708498678718, −2.12850911439626243136649876846, −1.05869337016024473579917875154, −1.03706032957268335909723535898, −0.66988130888711055423948482613, −0.43011152148520765474455225986,
0.43011152148520765474455225986, 0.66988130888711055423948482613, 1.03706032957268335909723535898, 1.05869337016024473579917875154, 2.12850911439626243136649876846, 2.15029142113237372708498678718, 2.49134454880760851150475675657, 2.63184626224489193992625927549, 3.35692932381284964919021160274, 3.51724645317691727385229881191, 3.53341609748013895997648511450, 3.68947910265967532936563385404, 4.03489177231261257232370131593, 4.06002108962734619602669889331, 4.49937841827387046102345269952, 4.57042721074303415721021521765, 5.56328869754516015083940980932, 5.61388162320183071262403076263, 5.70296552696148730013076207314, 5.83897559115442374706069078168, 5.99940538329425796228326822905, 6.34545777139186181357778014479, 6.40016633293207713285343821173, 6.48798977425343006227048134791, 6.90669347494207045335594061458
