| L(s) = 1 | − 2·4-s − 7-s + 3·16-s − 2·25-s + 2·28-s + 3·31-s + 2·37-s + 3·43-s + 49-s − 4·64-s + 2·67-s + 2·73-s + 3·79-s + 3·97-s + 4·100-s − 3·109-s − 3·112-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·4-s − 7-s + 3·16-s − 2·25-s + 2·28-s + 3·31-s + 2·37-s + 3·43-s + 49-s − 4·64-s + 2·67-s + 2·73-s + 3·79-s + 3·97-s + 4·100-s − 3·109-s − 3·112-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8982009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8982009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7618538460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7618538460\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 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
−9.154271191975965598519953132868, −8.916959237482519387056310559833, −8.157280186571733007295242923937, −8.133703344865286189584713406591, −7.71980973959988714175851047142, −7.55603787445622886040022175431, −6.57775274819716627244593393098, −6.44258114976702968098175906078, −6.01503434956038746304451939140, −5.68762939159491474891674745736, −5.21037268879840031033985225004, −4.81279238860985021866759334765, −4.19949990049989494500563973877, −4.19112232180308627983247014933, −3.67052226743233514489463443934, −3.28462014290239288790873072591, −2.49980844782213457072245957126, −2.32147515691755373219052352654, −0.917071028890371454801269321322, −0.798714747933958883067703284231,
0.798714747933958883067703284231, 0.917071028890371454801269321322, 2.32147515691755373219052352654, 2.49980844782213457072245957126, 3.28462014290239288790873072591, 3.67052226743233514489463443934, 4.19112232180308627983247014933, 4.19949990049989494500563973877, 4.81279238860985021866759334765, 5.21037268879840031033985225004, 5.68762939159491474891674745736, 6.01503434956038746304451939140, 6.44258114976702968098175906078, 6.57775274819716627244593393098, 7.55603787445622886040022175431, 7.71980973959988714175851047142, 8.133703344865286189584713406591, 8.157280186571733007295242923937, 8.916959237482519387056310559833, 9.154271191975965598519953132868
