| L(s) = 1 | − 16-s + 2·25-s − 4·67-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-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 | − 16-s + 2·25-s − 4·67-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7081386246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7081386246\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| 71 | $C_2^2$ | \( 1 + T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.75768382820734119807401977288, −10.96857346074114353539039441239, −10.58591257591691448054745128864, −10.51651999613889885513362405791, −9.637006426297288684903192565796, −9.329485489699679464914913646392, −8.737212900068718365040170104805, −8.669173095535633182753122905473, −7.82066734516852205356545624700, −7.44273120123202670099339716443, −6.96600112572225974598710138211, −6.28449274811526203539564201486, −6.21468501338760857365001128365, −5.07896495965005685939838322715, −5.01233410038919233840908818842, −4.31225387507632891502908836307, −3.64577065757865264279999126722, −2.91671596863141148818752592879, −2.36438985690773399053883083814, −1.32837335488096312434276989129,
1.32837335488096312434276989129, 2.36438985690773399053883083814, 2.91671596863141148818752592879, 3.64577065757865264279999126722, 4.31225387507632891502908836307, 5.01233410038919233840908818842, 5.07896495965005685939838322715, 6.21468501338760857365001128365, 6.28449274811526203539564201486, 6.96600112572225974598710138211, 7.44273120123202670099339716443, 7.82066734516852205356545624700, 8.669173095535633182753122905473, 8.737212900068718365040170104805, 9.329485489699679464914913646392, 9.637006426297288684903192565796, 10.51651999613889885513362405791, 10.58591257591691448054745128864, 10.96857346074114353539039441239, 11.75768382820734119807401977288
