| L(s) = 1 | + 6·4-s + 90·7-s + 2.09e3·13-s − 988·16-s + 2.31e3·19-s + 540·28-s + 7.26e3·31-s + 6.22e3·37-s − 2.07e4·43-s − 2.75e4·49-s + 1.25e4·52-s − 1.52e4·61-s − 1.20e4·64-s + 1.00e5·67-s + 1.49e5·73-s + 1.39e4·76-s + 7.66e4·79-s + 1.88e5·91-s − 1.43e5·97-s − 6.76e4·103-s − 2.28e5·109-s − 8.89e4·112-s − 2.10e5·121-s + 4.35e4·124-s + 127-s + 131-s + 2.08e5·133-s + ⋯ |
| L(s) = 1 | + 3/16·4-s + 0.694·7-s + 3.42·13-s − 0.964·16-s + 1.47·19-s + 0.130·28-s + 1.35·31-s + 0.746·37-s − 1.70·43-s − 1.63·49-s + 0.643·52-s − 0.523·61-s − 0.368·64-s + 2.74·67-s + 3.28·73-s + 0.276·76-s + 1.38·79-s + 2.38·91-s − 1.54·97-s − 0.628·103-s − 1.83·109-s − 0.669·112-s − 1.30·121-s + 0.254·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 1.02·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.938868821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.938868821\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 45 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 210102 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 1045 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1222434 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 61 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1682834 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27470298 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3633 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3110 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 82895598 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10355 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 275888094 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 736930506 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 601496598 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7613 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 50445 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 2896613298 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 74710 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 38316 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6600582406 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3347081102 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 71755 T + p^{5} T^{2} )^{2} \) |
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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
−11.39780710074477813358412381856, −11.06468846413703353678553410727, −11.05256584021755537439970437189, −10.24366679856586516772470812104, −9.466668680882592657590225982682, −9.332508103995672890747280677021, −8.416056210735920930563230592384, −8.143086240788727262840998203037, −8.016539638538656105296084977745, −6.81872810496914990776812746302, −6.56462768505491543374968463401, −6.12688199683537207554657381495, −5.31723602065421553904255484082, −4.91583179100084402747218125644, −3.99521338835471761866060929792, −3.59649200054271598013008780809, −2.92499233562468193415137611591, −1.88708991751839917380218294907, −1.25012765400681611986288970247, −0.75255882818596489184915003534,
0.75255882818596489184915003534, 1.25012765400681611986288970247, 1.88708991751839917380218294907, 2.92499233562468193415137611591, 3.59649200054271598013008780809, 3.99521338835471761866060929792, 4.91583179100084402747218125644, 5.31723602065421553904255484082, 6.12688199683537207554657381495, 6.56462768505491543374968463401, 6.81872810496914990776812746302, 8.016539638538656105296084977745, 8.143086240788727262840998203037, 8.416056210735920930563230592384, 9.332508103995672890747280677021, 9.466668680882592657590225982682, 10.24366679856586516772470812104, 11.05256584021755537439970437189, 11.06468846413703353678553410727, 11.39780710074477813358412381856
