| L(s) = 1 | + 12·7-s + 50·9-s − 60·17-s + 356·23-s − 25·25-s − 40·31-s + 500·41-s − 428·47-s − 578·49-s + 600·63-s + 200·71-s + 460·73-s + 2.64e3·79-s + 1.77e3·81-s − 1.74e3·89-s − 620·97-s + 2.80e3·103-s − 3.02e3·113-s − 720·119-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.00e3·153-s + ⋯ |
| L(s) = 1 | + 0.647·7-s + 1.85·9-s − 0.856·17-s + 3.22·23-s − 1/5·25-s − 0.231·31-s + 1.90·41-s − 1.32·47-s − 1.68·49-s + 1.19·63-s + 0.334·71-s + 0.737·73-s + 3.75·79-s + 2.42·81-s − 2.08·89-s − 0.648·97-s + 2.68·103-s − 2.51·113-s − 0.554·119-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 1.58·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.655432611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.655432611\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 938 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1894 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12118 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 178 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 21222 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 101206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 138850 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 57654 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 229242 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 391462 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2450 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1320 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 179250 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 874 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 310 T + p^{3} T^{2} )^{2} \) |
| 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
−9.488545604752342363081879971220, −9.131089968054584979756754053159, −8.874187213699615644288686830090, −8.129790602217912885143406274182, −7.83814030334685579067299963198, −7.53436504593675575672507494707, −6.86357055416189943977394512739, −6.78146669340449770858592991616, −6.43785196334441514812062745085, −5.68051938983478884997798942980, −4.99512330486873872905173745811, −4.79012015147302774856233374044, −4.64680339207890780948939755518, −3.72450453715078031695408496850, −3.63602816428242171539743033320, −2.69988355327150261673269357363, −2.31716885589250679000601961940, −1.41265709925717057253174356607, −1.29823691443167736644770991193, −0.53820478406691833161297783093,
0.53820478406691833161297783093, 1.29823691443167736644770991193, 1.41265709925717057253174356607, 2.31716885589250679000601961940, 2.69988355327150261673269357363, 3.63602816428242171539743033320, 3.72450453715078031695408496850, 4.64680339207890780948939755518, 4.79012015147302774856233374044, 4.99512330486873872905173745811, 5.68051938983478884997798942980, 6.43785196334441514812062745085, 6.78146669340449770858592991616, 6.86357055416189943977394512739, 7.53436504593675575672507494707, 7.83814030334685579067299963198, 8.129790602217912885143406274182, 8.874187213699615644288686830090, 9.131089968054584979756754053159, 9.488545604752342363081879971220
