| L(s) = 1 | + 10·7-s − 3·9-s − 12·19-s + 7·25-s + 6·31-s + 4·37-s + 32·43-s + 61·49-s − 30·63-s + 4·67-s + 24·73-s − 2·79-s + 12·103-s + 4·109-s − 13·121-s + 127-s + 131-s − 120·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 36·171-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 9-s − 2.75·19-s + 7/5·25-s + 1.07·31-s + 0.657·37-s + 4.87·43-s + 61/7·49-s − 3.77·63-s + 0.488·67-s + 2.80·73-s − 0.225·79-s + 1.18·103-s + 0.383·109-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s − 10.4·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 2.75·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.472531192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.472531192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ah_a_y |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_n_a_bw |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abc_a_uo |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aw_a_hn |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) | 4.19.m_du_xc_enj |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_k_a_aqn |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adu_a_gbb |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.31.ag_cz_apa_fjw |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_ack_aq_gcp |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_cq_a_gru |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.43.abg_vk_ajdo_ctik |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abu_a_adp |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_z_a_adga |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_ael_a_oku |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_es_a_qnj |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ae_aes_aq_typ |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_e_a_oxy |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ay_ow_agxc_cqnr |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.c_afz_c_bbse |
| 83 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ha_a_bgql |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_acs_a_aemf |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) | 4.97.a_amw_a_crcl |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.434433031687612729550020157324, −8.086120145241144515194198703059, −8.067934343432161241054448447716, −7.64455764971259306105257355115, −7.59966122631732781563735896525, −7.34262224233321903882717503206, −6.71044161535958250028334164301, −6.70890832276067215807618294663, −6.33635423387891132184298810170, −6.00076783053197475316371868028, −5.63666079382176366352863989216, −5.50529876586778317533940923165, −5.28755084966074291581227900464, −4.77255271040090010641930647368, −4.56842575147709646841048389374, −4.52399597871888444051520260104, −4.20633735695541841148947083885, −3.89850917740199073788705009239, −3.50522863587242324256290191617, −2.60830990057005144687847213990, −2.44087620048846942370061215593, −2.37458407069019780200715189050, −1.96256300809264731868526966723, −1.07558526971039686071547525448, −1.04784546751905315534518203223,
1.04784546751905315534518203223, 1.07558526971039686071547525448, 1.96256300809264731868526966723, 2.37458407069019780200715189050, 2.44087620048846942370061215593, 2.60830990057005144687847213990, 3.50522863587242324256290191617, 3.89850917740199073788705009239, 4.20633735695541841148947083885, 4.52399597871888444051520260104, 4.56842575147709646841048389374, 4.77255271040090010641930647368, 5.28755084966074291581227900464, 5.50529876586778317533940923165, 5.63666079382176366352863989216, 6.00076783053197475316371868028, 6.33635423387891132184298810170, 6.70890832276067215807618294663, 6.71044161535958250028334164301, 7.34262224233321903882717503206, 7.59966122631732781563735896525, 7.64455764971259306105257355115, 8.067934343432161241054448447716, 8.086120145241144515194198703059, 8.434433031687612729550020157324
