| L(s) = 1 | + 2·3-s + 12·7-s − 5·9-s − 20·13-s + 4·19-s + 24·21-s + 18·25-s − 28·27-s + 44·31-s + 12·37-s − 40·39-s + 164·43-s + 10·49-s + 8·57-s + 172·61-s − 60·63-s + 4·67-s + 164·73-s + 36·75-s − 20·79-s − 11·81-s − 240·91-s + 88·93-s − 188·97-s + 268·103-s − 20·109-s + 24·111-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 12/7·7-s − 5/9·9-s − 1.53·13-s + 4/19·19-s + 8/7·21-s + 0.719·25-s − 1.03·27-s + 1.41·31-s + 0.324·37-s − 1.02·39-s + 3.81·43-s + 0.204·49-s + 8/57·57-s + 2.81·61-s − 0.952·63-s + 4/67·67-s + 2.24·73-s + 0.479·75-s − 0.253·79-s − 0.135·81-s − 2.63·91-s + 0.946·93-s − 1.93·97-s + 2.60·103-s − 0.183·109-s + 8/37·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.691447781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691447781\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 66 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 930 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1746 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1554 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5406 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8370 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14690 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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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
−12.52270550173641442467250105425, −12.07533194086664122703912395565, −11.46749126503602838245904584622, −11.15273418806810480033068710746, −10.77243614813820501516033385695, −9.807868415336828821423410667152, −9.751796755197434385074848354339, −8.860349239253088693888270816133, −8.575045983740774084667128233337, −7.895983195278810667827954871521, −7.68884831408676266610086175556, −7.11015658890692328733828258675, −6.28255053067563499979673230672, −5.50803362391592836641157649931, −4.97674300282699422541812807906, −4.51932106171602055780806225932, −3.71510029959258957485593944174, −2.44784232092364310126463474209, −2.44172152162155152725027068676, −0.979802016761469266475686427400,
0.979802016761469266475686427400, 2.44172152162155152725027068676, 2.44784232092364310126463474209, 3.71510029959258957485593944174, 4.51932106171602055780806225932, 4.97674300282699422541812807906, 5.50803362391592836641157649931, 6.28255053067563499979673230672, 7.11015658890692328733828258675, 7.68884831408676266610086175556, 7.895983195278810667827954871521, 8.575045983740774084667128233337, 8.860349239253088693888270816133, 9.751796755197434385074848354339, 9.807868415336828821423410667152, 10.77243614813820501516033385695, 11.15273418806810480033068710746, 11.46749126503602838245904584622, 12.07533194086664122703912395565, 12.52270550173641442467250105425
