| L(s) = 1 | + 2·3-s + 3·9-s + 2·19-s − 2·25-s + 4·27-s − 2·31-s + 2·37-s + 4·57-s − 4·75-s + 5·81-s − 4·93-s − 2·103-s − 2·109-s + 4·111-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 6·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2·19-s − 2·25-s + 4·27-s − 2·31-s + 2·37-s + 4·57-s − 4·75-s + 5·81-s − 4·93-s − 2·103-s − 2·109-s + 4·111-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 6·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.023035355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.023035355\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.290482101164245452291464194591, −9.222062084050737052921673004828, −8.542173734587199270846370966433, −8.093992730379442651950593693811, −7.88251214804175251153751013057, −7.62559400836656336361540568784, −7.12015646167438576960753738447, −7.06365802641872893874730740151, −6.24842367342337009154460249183, −5.94431376429737364626557300661, −5.19698848139289390430207371670, −5.14581485904097073347013384649, −4.16479207280695905422706500178, −4.11459919652790620964795266808, −3.57739591976688341342433447908, −3.25750276778918626683902306310, −2.56947549379523153126500478508, −2.39580999772606537566726123613, −1.58309807623782885716503054670, −1.24286079441251207481466677066,
1.24286079441251207481466677066, 1.58309807623782885716503054670, 2.39580999772606537566726123613, 2.56947549379523153126500478508, 3.25750276778918626683902306310, 3.57739591976688341342433447908, 4.11459919652790620964795266808, 4.16479207280695905422706500178, 5.14581485904097073347013384649, 5.19698848139289390430207371670, 5.94431376429737364626557300661, 6.24842367342337009154460249183, 7.06365802641872893874730740151, 7.12015646167438576960753738447, 7.62559400836656336361540568784, 7.88251214804175251153751013057, 8.093992730379442651950593693811, 8.542173734587199270846370966433, 9.222062084050737052921673004828, 9.290482101164245452291464194591
