| L(s) = 1 | − 9-s + 8·13-s − 6·17-s − 8·29-s + 16·37-s + 10·41-s − 14·49-s + 8·53-s − 16·61-s − 18·73-s − 8·81-s + 30·89-s − 4·97-s − 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.21·13-s − 1.45·17-s − 1.48·29-s + 2.63·37-s + 1.56·41-s − 2·49-s + 1.09·53-s − 2.04·61-s − 2.10·73-s − 8/9·81-s + 3.17·89-s − 0.406·97-s − 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.529531312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529531312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.187017328247638614254152943637, −7.87947007029256221948434742723, −7.47936130293416306525307192240, −7.30177669724537553505152341655, −6.62940560131177360151873950474, −6.30048662820212128597592051853, −6.12895175587398555693508087575, −5.91076880022242422181209497171, −5.45350957374437843672198243034, −4.98182644668414629674938856630, −4.34586177095589575416896267621, −4.27492176168407701402275717824, −3.95778197531840804477655444328, −3.36129759813884846978934055298, −2.97582974023882825779574740362, −2.65467377156982528207715161233, −1.90725591508502306000371935443, −1.69816137559948652563091035093, −1.00602415346545784953591029377, −0.44863285696777878211461281221,
0.44863285696777878211461281221, 1.00602415346545784953591029377, 1.69816137559948652563091035093, 1.90725591508502306000371935443, 2.65467377156982528207715161233, 2.97582974023882825779574740362, 3.36129759813884846978934055298, 3.95778197531840804477655444328, 4.27492176168407701402275717824, 4.34586177095589575416896267621, 4.98182644668414629674938856630, 5.45350957374437843672198243034, 5.91076880022242422181209497171, 6.12895175587398555693508087575, 6.30048662820212128597592051853, 6.62940560131177360151873950474, 7.30177669724537553505152341655, 7.47936130293416306525307192240, 7.87947007029256221948434742723, 8.187017328247638614254152943637
