| L(s) = 1 | − 28·7-s − 44·11-s + 100·25-s + 490·49-s + 1.23e3·77-s + 146·81-s − 152·107-s + 328·113-s + 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2.80e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4·7-s − 4·11-s + 4·25-s + 10·49-s + 16·77-s + 1.80·81-s − 1.42·107-s + 2.90·113-s + 10·121-s + 0.00787·127-s + 0.00763·131-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 + 0.00578·173-s − 16·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7821588021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7821588021\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 146 T^{4} + p^{8} T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 12178 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 142094 T^{4} + p^{8} T^{8} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 1378574 T^{4} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2255822 T^{4} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 926 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 6675826 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2846 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 13711186 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 12099182 T^{4} + p^{8} T^{8} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 56328014 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 12466 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.74548712673059782735823056768, −6.45633757927332486480338454369, −6.32192413249289585298779440304, −6.25537441630081195133523786954, −5.91772654043961330488388394215, −5.51651462152505430860787463868, −5.44392088276792494440854671487, −5.14120241212597921199381859200, −5.12951678516407708420992873905, −4.86801870634779303582481746827, −4.44380132891713258463472375779, −4.21597043107844160179694019987, −3.89955662496109751248162859781, −3.51157028426922835478801756720, −3.18852503001976803303757302774, −3.17385597132120420944542159349, −3.02590411113083116099458247001, −2.58368150229558019645112819919, −2.55035467324904014736429444168, −2.47013344894499896326384720482, −1.93992981350260456977573418676, −1.18068152680698404969880027701, −0.69987523740431655165079512414, −0.50927860443159417301888213036, −0.24680139826550981414275168042,
0.24680139826550981414275168042, 0.50927860443159417301888213036, 0.69987523740431655165079512414, 1.18068152680698404969880027701, 1.93992981350260456977573418676, 2.47013344894499896326384720482, 2.55035467324904014736429444168, 2.58368150229558019645112819919, 3.02590411113083116099458247001, 3.17385597132120420944542159349, 3.18852503001976803303757302774, 3.51157028426922835478801756720, 3.89955662496109751248162859781, 4.21597043107844160179694019987, 4.44380132891713258463472375779, 4.86801870634779303582481746827, 5.12951678516407708420992873905, 5.14120241212597921199381859200, 5.44392088276792494440854671487, 5.51651462152505430860787463868, 5.91772654043961330488388394215, 6.25537441630081195133523786954, 6.32192413249289585298779440304, 6.45633757927332486480338454369, 6.74548712673059782735823056768
