L(s) = 1 | + 8·61-s − 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + ⋯ |
L(s) = 1 | + 8·61-s − 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.043788109\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.043788109\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$ | \( ( 1 - T )^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−7.09667085380094546767123062037, −6.80058199737320817471580939049, −6.74884801600807357590526138135, −6.65009205907508451557026934974, −6.57738372855317584120674119835, −5.92011309673505568806087375931, −5.72985770508222206291467081173, −5.69567846077450316822158419440, −5.43690937806494213244648617856, −5.12146938351550802448656829725, −5.07473701428459983573087400675, −4.76270548427746030080657825137, −4.35211579153683385784979435733, −4.12131701679073215332867686006, −3.90709732369647258909861137397, −3.81653063637242672878343513336, −3.52784041319034434727517903042, −3.01609200972239756683414937444, −2.98876520402881796355534311222, −2.47595419092273291458371071096, −2.28492571135972876086159921573, −2.07364776671274661445798946880, −1.67850746664169899658497711978, −1.03672170738375410860717161310, −0.874279031667763229495285711211,
0.874279031667763229495285711211, 1.03672170738375410860717161310, 1.67850746664169899658497711978, 2.07364776671274661445798946880, 2.28492571135972876086159921573, 2.47595419092273291458371071096, 2.98876520402881796355534311222, 3.01609200972239756683414937444, 3.52784041319034434727517903042, 3.81653063637242672878343513336, 3.90709732369647258909861137397, 4.12131701679073215332867686006, 4.35211579153683385784979435733, 4.76270548427746030080657825137, 5.07473701428459983573087400675, 5.12146938351550802448656829725, 5.43690937806494213244648617856, 5.69567846077450316822158419440, 5.72985770508222206291467081173, 5.92011309673505568806087375931, 6.57738372855317584120674119835, 6.65009205907508451557026934974, 6.74884801600807357590526138135, 6.80058199737320817471580939049, 7.09667085380094546767123062037