| L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 2·7-s − 2·8-s − 4·10-s + 4·11-s − 4·13-s + 4·14-s − 4·16-s + 6·19-s − 2·20-s + 8·22-s − 16·23-s + 11·25-s − 8·26-s + 2·28-s + 12·29-s + 4·31-s − 2·32-s − 4·35-s − 4·37-s + 12·38-s + 4·40-s − 16·41-s − 12·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s − 0.707·8-s − 1.26·10-s + 1.20·11-s − 1.10·13-s + 1.06·14-s − 16-s + 1.37·19-s − 0.447·20-s + 1.70·22-s − 3.33·23-s + 11/5·25-s − 1.56·26-s + 0.377·28-s + 2.22·29-s + 0.718·31-s − 0.353·32-s − 0.676·35-s − 0.657·37-s + 1.94·38-s + 0.632·40-s − 2.49·41-s − 1.82·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.902190414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.902190414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.c_ah_c_cy |
| 7 | $D_4\times C_2$ | \( 1 - 2 T + 20 T^{3} - 61 T^{4} + 20 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ac_a_u_acj |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ae_bc_ado_sg |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) | 4.13.e_ao_q_ud |
| 17 | $C_2^3$ | \( 1 - 23 T^{2} + 240 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ax_a_jg |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 12 T^{3} + 251 T^{4} + 12 p T^{5} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_a_m_jr |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.23.q_hg_cai_lpu |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_de_aya_gjj |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ae_ee_amu_hfm |
| 41 | $D_4\times C_2$ | \( 1 + 16 T + 121 T^{2} + 848 T^{3} + 6048 T^{4} + 848 p T^{5} + 121 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.q_er_bgq_iyq |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 84 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.m_hw_cgq_ums |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.e_gq_ue_rnu |
| 53 | $D_4\times C_2$ | \( 1 - 4 T - 50 T^{2} + 160 T^{3} + 699 T^{4} + 160 p T^{5} - 50 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ae_aby_ge_bax |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 62 T^{2} + 160 T^{3} + 1659 T^{4} + 160 p T^{5} - 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ae_ack_ge_clv |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 63 T^{2} - 40 T^{3} + 8504 T^{4} - 40 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ai_acl_abo_mpc |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 96 T^{2} - 12 T^{3} + 10523 T^{4} - 12 p T^{5} - 96 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.g_ads_am_pot |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 104 T^{2} + 12 T^{3} + 12003 T^{4} + 12 p T^{5} - 104 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ag_aea_m_rtr |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ay_qe_ahmu_cxba |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 32 T^{2} + 540 T^{3} - 4461 T^{4} + 540 p T^{5} - 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ag_abg_uu_agpp |
| 83 | $D_4\times C_2$ | \( 1 - 4 T - 110 T^{2} + 160 T^{3} + 7659 T^{4} + 160 p T^{5} - 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ae_aeg_ge_lip |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 59 T^{2} + 300 T^{3} + 20472 T^{4} + 300 p T^{5} - 59 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.m_ach_lo_behk |
| 97 | $D_{4}$ | \( ( 1 - 14 T + 199 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abc_ww_amgu_fmrj |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52342134549982753598758735900, −7.34099306845876720174753007870, −6.94886536086931612280175369062, −6.70124185730592491019419282185, −6.52194098258030178554266524651, −6.48474844119719248625710136694, −6.38022474132996650544428413706, −5.59778466146541214370119250160, −5.59713490457440970944396125092, −5.34922967743570958637662526372, −4.95093767439934545922267358416, −4.93104153534609319858582970541, −4.68798763830789908906455402614, −4.32925456134807158443272191149, −4.25568231709419747125195768785, −3.80168941827729583107200529715, −3.46999258393496830350935504462, −3.42229522551384537547121727666, −3.28258758548054647280679636605, −2.56359744734095642705544406438, −2.38105576694336703510226794493, −2.13806388583065570767034494071, −1.31524018983352230055320536149, −1.29089172130969170223422834849, −0.39168328150696687332452967169,
0.39168328150696687332452967169, 1.29089172130969170223422834849, 1.31524018983352230055320536149, 2.13806388583065570767034494071, 2.38105576694336703510226794493, 2.56359744734095642705544406438, 3.28258758548054647280679636605, 3.42229522551384537547121727666, 3.46999258393496830350935504462, 3.80168941827729583107200529715, 4.25568231709419747125195768785, 4.32925456134807158443272191149, 4.68798763830789908906455402614, 4.93104153534609319858582970541, 4.95093767439934545922267358416, 5.34922967743570958637662526372, 5.59713490457440970944396125092, 5.59778466146541214370119250160, 6.38022474132996650544428413706, 6.48474844119719248625710136694, 6.52194098258030178554266524651, 6.70124185730592491019419282185, 6.94886536086931612280175369062, 7.34099306845876720174753007870, 7.52342134549982753598758735900
