| L(s) = 1 | + 2·4-s − 20·11-s + 3·16-s − 8·19-s − 12·29-s − 12·31-s − 40·44-s + 22·49-s + 36·59-s + 4·61-s + 12·64-s − 8·71-s − 16·76-s + 24·79-s + 4·101-s + 32·109-s − 24·116-s + 210·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4-s − 6.03·11-s + 3/4·16-s − 1.83·19-s − 2.22·29-s − 2.15·31-s − 6.03·44-s + 22/7·49-s + 4.68·59-s + 0.512·61-s + 3/2·64-s − 0.949·71-s − 1.83·76-s + 2.70·79-s + 0.398·101-s + 3.06·109-s − 2.22·116-s + 19.0·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063164036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063164036\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.2.a_ac_a_b |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_aw_a_id |
| 11 | $D_{4}$ | \( ( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.u_hi_brc_gpv |
| 17 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1171 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aby_a_btb |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.i_bk_iy_cda |
| 23 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ae_a_adm |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.m_fy_boq_ksx |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.m_fm_bmu_khv |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_acy_a_gew |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_dw_a_irm |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_afs_a_njy |
| 47 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 8707 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_afe_a_mwx |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahm_a_wfz |
| 59 | $D_{4}$ | \( ( 1 - 18 T + 181 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.abk_bak_amui_elih |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.61.ae_jq_abci_bicx |
| 67 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 9939 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) | 4.67.a_afe_a_osh |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.i_ca_bbk_stq |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) | 4.73.a_aim_a_bhri |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.79.ay_um_ajsa_edus |
| 83 | $D_4\times C_2$ | \( 1 - 118 T^{2} + 13731 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aeo_a_uid |
| 89 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ie_a_bobm |
| 97 | $D_4\times C_2$ | \( 1 - 364 T^{2} + 51814 T^{4} - 364 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_aoa_a_cyqw |
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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
−6.01156856157052536703001198748, −5.88133648216457303380356549454, −5.76801009423955381661991631722, −5.57072754006188578476772568107, −5.35311427129945455502704853259, −5.31749704677881817972305373803, −5.15527496094235511277281412854, −4.83190338911288352018573193221, −4.73814914539608412929311329936, −4.39038684146317382678329003968, −4.09601829384017909706061652543, −3.70963507536633233598538881534, −3.67931374613433992499760682635, −3.38739930268384852964131204809, −3.32145636935147308205534175539, −2.58613182662098985776943457755, −2.56851517990264228619921251063, −2.56438560239327494346287948589, −2.28399189529141503760130614661, −2.08857770246255187522710336611, −2.00591791841107213128912588380, −1.62786883785073042423797485197, −0.68528087287477569589223603097, −0.62574131222291821066748274744, −0.22821993213323180312421578222,
0.22821993213323180312421578222, 0.62574131222291821066748274744, 0.68528087287477569589223603097, 1.62786883785073042423797485197, 2.00591791841107213128912588380, 2.08857770246255187522710336611, 2.28399189529141503760130614661, 2.56438560239327494346287948589, 2.56851517990264228619921251063, 2.58613182662098985776943457755, 3.32145636935147308205534175539, 3.38739930268384852964131204809, 3.67931374613433992499760682635, 3.70963507536633233598538881534, 4.09601829384017909706061652543, 4.39038684146317382678329003968, 4.73814914539608412929311329936, 4.83190338911288352018573193221, 5.15527496094235511277281412854, 5.31749704677881817972305373803, 5.35311427129945455502704853259, 5.57072754006188578476772568107, 5.76801009423955381661991631722, 5.88133648216457303380356549454, 6.01156856157052536703001198748
