| L(s) = 1 | − 2·4-s − 2·7-s − 4·9-s + 2·13-s − 6·19-s + 6·23-s + 4·28-s + 6·29-s − 2·31-s + 8·36-s + 4·37-s + 12·41-s + 10·43-s − 6·47-s + 3·49-s − 4·52-s + 6·53-s + 12·59-s + 12·61-s + 8·63-s + 8·64-s + 12·67-s − 12·71-s + 10·73-s + 12·76-s + 14·79-s + 7·81-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 4/3·9-s + 0.554·13-s − 1.37·19-s + 1.25·23-s + 0.755·28-s + 1.11·29-s − 0.359·31-s + 4/3·36-s + 0.657·37-s + 1.87·41-s + 1.52·43-s − 0.875·47-s + 3/7·49-s − 0.554·52-s + 0.824·53-s + 1.56·59-s + 1.53·61-s + 1.00·63-s + 64-s + 1.46·67-s − 1.42·71-s + 1.17·73-s + 1.37·76-s + 1.57·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.502491487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502491487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 128 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 153 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 185 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.937024426524079144787328360258, −8.910406033472334021761878303607, −8.518212930767048461625161348965, −8.330868310818645124083279105687, −7.78766878529857403449601353921, −7.12573819515663456994728866401, −7.00441240009104713526450222242, −6.33025607128642050639931042922, −5.97047985237160563130376249014, −5.90297176870148273056343193429, −5.08512229048410912864508888942, −4.96643189759817778648104500327, −4.29116339999626558377485545147, −3.99286513803971456673772993945, −3.51991001446845556787659807396, −2.99212890958828416945644830971, −2.44134627710096431805781094842, −2.16686317087043060142253087152, −0.72276881933892663336381580117, −0.69143204985789991160848992789,
0.69143204985789991160848992789, 0.72276881933892663336381580117, 2.16686317087043060142253087152, 2.44134627710096431805781094842, 2.99212890958828416945644830971, 3.51991001446845556787659807396, 3.99286513803971456673772993945, 4.29116339999626558377485545147, 4.96643189759817778648104500327, 5.08512229048410912864508888942, 5.90297176870148273056343193429, 5.97047985237160563130376249014, 6.33025607128642050639931042922, 7.00441240009104713526450222242, 7.12573819515663456994728866401, 7.78766878529857403449601353921, 8.330868310818645124083279105687, 8.518212930767048461625161348965, 8.910406033472334021761878303607, 8.937024426524079144787328360258
