| L(s) = 1 | + 16·17-s − 8·25-s + 16·41-s − 24·49-s − 16·73-s + 64·89-s + 24·97-s + 32·113-s − 20·121-s + ⋯ |
| L(s) = 1 | + 3.88·17-s − 8/5·25-s + 2.49·41-s − 3.42·49-s − 1.87·73-s + 6.78·89-s + 2.43·97-s + 3.01·113-s − 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.141952539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.141952539\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_i_a_co |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_y_a_ji |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_u_a_ne |
| 13 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_e_a_ne |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.17.aq_gi_abpg_huc |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_au_a_bfq |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bc_a_bwg |
| 29 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ea_a_gms |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_eq_a_iek |
| 37 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_dw_a_htm |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.41.aq_ka_adho_bayo |
| 43 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_cy_a_hpu |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_eu_a_mfu |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ea_a_mic |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_afs_a_skk |
| 61 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ho_a_zfq |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.67.a_ki_a_bnvy |
| 71 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_bc_a_pfi |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.73.q_oy_foq_cqks |
| 79 | $C_2^2$ | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ii_a_bjss |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_em_a_zji |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) | 4.89.acm_cuu_abxnk_vzsg |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.97.ay_xg_alpw_frkw |
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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
−5.99099403495706668955644562406, −5.75847813225027814418201801914, −5.44897765612470604599926325530, −5.39964650025734038334357711465, −5.05255637025177608125188705439, −4.79909383224258825004834430031, −4.75628857675733656447259505000, −4.59351529206510630464184421610, −4.43220427854209321098822670179, −3.88075901163517833051891350768, −3.75650517247086482236838000980, −3.60831019286873278136412371419, −3.59185758374375663150232920207, −3.20853513645789752437782973557, −3.03213850573840251631017500889, −2.88199546820798744855326852443, −2.73750185431110382328593866208, −2.13011582929890769845634910224, −1.94613753697404705061423507002, −1.80922908077163399303214316143, −1.73437906404678433804948375584, −1.04727951821994433156680124642, −0.961092984726175068772015173827, −0.73970149342367046213149823429, −0.35521274087390964463913027938,
0.35521274087390964463913027938, 0.73970149342367046213149823429, 0.961092984726175068772015173827, 1.04727951821994433156680124642, 1.73437906404678433804948375584, 1.80922908077163399303214316143, 1.94613753697404705061423507002, 2.13011582929890769845634910224, 2.73750185431110382328593866208, 2.88199546820798744855326852443, 3.03213850573840251631017500889, 3.20853513645789752437782973557, 3.59185758374375663150232920207, 3.60831019286873278136412371419, 3.75650517247086482236838000980, 3.88075901163517833051891350768, 4.43220427854209321098822670179, 4.59351529206510630464184421610, 4.75628857675733656447259505000, 4.79909383224258825004834430031, 5.05255637025177608125188705439, 5.39964650025734038334357711465, 5.44897765612470604599926325530, 5.75847813225027814418201801914, 5.99099403495706668955644562406
