| L(s) = 1 | + 4·3-s + 8·9-s + 4·11-s + 7·16-s + 4·23-s + 20·27-s + 8·31-s + 16·33-s − 12·37-s − 12·47-s + 28·48-s + 4·53-s − 12·67-s + 16·69-s − 32·71-s + 50·81-s + 32·93-s + 28·97-s + 32·99-s − 36·103-s − 48·111-s − 36·113-s − 10·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 1.20·11-s + 7/4·16-s + 0.834·23-s + 3.84·27-s + 1.43·31-s + 2.78·33-s − 1.97·37-s − 1.75·47-s + 4.04·48-s + 0.549·53-s − 1.46·67-s + 1.92·69-s − 3.79·71-s + 50/9·81-s + 3.31·93-s + 2.84·97-s + 3.21·99-s − 3.54·103-s − 4.55·111-s − 3.38·113-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.390518403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.390518403\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) | 4.2.a_a_a_ah |
| 3 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ae_i_au_bu |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.7.a_a_a_du |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) | 4.13.a_a_a_alq |
| 17 | $C_2^3$ | \( 1 - 382 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_aos |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ae_a_bby |
| 23 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_i_adw_bvy |
| 29 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bk_a_cze |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.31.ai_fs_abdw_ktq |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.m_cu_zk_img |
| 41 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_adg_a_hpe |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.43.a_a_a_fmg |
| 47 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.m_cu_bea_mao |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.ae_i_aim_iyo |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agi_a_ugk |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agi_a_uyw |
| 67 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_cu_bng_uxi |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.71.bg_zs_ncy_fbmc |
| 73 | $C_2^3$ | \( 1 + 5218 T^{4} + p^{4} T^{8} \) | 4.73.a_a_a_hss |
| 79 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_jc_a_bnbq |
| 83 | $C_2^3$ | \( 1 - 6382 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_ajlm |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abc_pc_aica_duhq |
| 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
−8.760360539341556557451499637835, −8.362567358667186050037969447665, −8.257816441447557133191724945344, −8.036013422726171460484632312321, −7.71551579849435041295105008814, −7.37581355270162389275710725808, −7.36827518574286363005329649422, −6.86150653250175145653782511441, −6.51835193074540311618730309542, −6.41459832462566557158604211094, −6.30355641629621467751872881346, −5.64613102019009415123944578493, −5.34008849130754469357084379456, −5.05703186062060911292846631153, −4.84234464089211019012546364583, −4.18644924423602865851075589669, −4.14021418918758867503301822080, −3.84685670890152207854793355263, −3.29893448568374250541462654086, −3.04326589036324223350605420569, −2.91490074568818504483938368078, −2.70535457588552396529025107515, −1.90241289215724974656017282080, −1.36153252467255927321820151422, −1.28410445897029585626401370469,
1.28410445897029585626401370469, 1.36153252467255927321820151422, 1.90241289215724974656017282080, 2.70535457588552396529025107515, 2.91490074568818504483938368078, 3.04326589036324223350605420569, 3.29893448568374250541462654086, 3.84685670890152207854793355263, 4.14021418918758867503301822080, 4.18644924423602865851075589669, 4.84234464089211019012546364583, 5.05703186062060911292846631153, 5.34008849130754469357084379456, 5.64613102019009415123944578493, 6.30355641629621467751872881346, 6.41459832462566557158604211094, 6.51835193074540311618730309542, 6.86150653250175145653782511441, 7.36827518574286363005329649422, 7.37581355270162389275710725808, 7.71551579849435041295105008814, 8.036013422726171460484632312321, 8.257816441447557133191724945344, 8.362567358667186050037969447665, 8.760360539341556557451499637835
