| L(s) = 1 | − 2·9-s + 3·81-s − 8·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 2·9-s + 3·81-s − 8·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6172872165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6172872165\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$ | \( ( 1 + T )^{8} \) |
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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
−6.79086485975441440948786605513, −6.73824302184038454308079115223, −6.70469351939790803109367351248, −6.06721125677170315531463711802, −6.02447825685202347727637113729, −6.01164636642245522598555766092, −5.68953661969341963097242831986, −5.32786036996171564093830852157, −5.30535955489511374760814749431, −5.15416656696291014008391082304, −4.76938348563243513634370399072, −4.52003089030410023500199749084, −4.19659847462567164862127246443, −4.14019518564257500800259096714, −3.77209691869147099429817644126, −3.38582658671152759664466590645, −3.33399801338009607358460732162, −3.05760073405233980040513067642, −2.59566197107718367984931429705, −2.48685273683211512943679477018, −2.47918794121677231774986272014, −1.83323515450409766242867810768, −1.50900546726229805037942865789, −1.21128091372142984625068142395, −0.47723753458551672793786135729,
0.47723753458551672793786135729, 1.21128091372142984625068142395, 1.50900546726229805037942865789, 1.83323515450409766242867810768, 2.47918794121677231774986272014, 2.48685273683211512943679477018, 2.59566197107718367984931429705, 3.05760073405233980040513067642, 3.33399801338009607358460732162, 3.38582658671152759664466590645, 3.77209691869147099429817644126, 4.14019518564257500800259096714, 4.19659847462567164862127246443, 4.52003089030410023500199749084, 4.76938348563243513634370399072, 5.15416656696291014008391082304, 5.30535955489511374760814749431, 5.32786036996171564093830852157, 5.68953661969341963097242831986, 6.01164636642245522598555766092, 6.02447825685202347727637113729, 6.06721125677170315531463711802, 6.70469351939790803109367351248, 6.73824302184038454308079115223, 6.79086485975441440948786605513
