| L(s) = 1 | + 2·3-s + 4·5-s − 3·7-s + 9-s + 3·11-s − 5·13-s + 8·15-s + 3·19-s − 6·21-s + 8·23-s + 14·25-s − 2·27-s − 14·29-s − 31-s + 6·33-s − 12·35-s − 9·37-s − 10·39-s + 6·41-s + 3·43-s + 4·45-s − 4·47-s + 6·49-s − 3·53-s + 12·55-s + 6·57-s − 11·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.38·13-s + 2.06·15-s + 0.688·19-s − 1.30·21-s + 1.66·23-s + 14/5·25-s − 0.384·27-s − 2.59·29-s − 0.179·31-s + 1.04·33-s − 2.02·35-s − 1.47·37-s − 1.60·39-s + 0.937·41-s + 0.457·43-s + 0.596·45-s − 0.583·47-s + 6/7·49-s − 0.412·53-s + 1.61·55-s + 0.794·57-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.301339272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.301339272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ae_c_aq_dn |
| 7 | $D_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 24 T^{3} - 76 T^{4} - 24 p T^{5} + 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.d_d_ay_acy |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - 19 T^{2} + p^{2} T^{4} ) \) | 4.11.ad_af_y_ay |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + 3 T^{2} - 20 T^{3} - 10 T^{4} - 20 p T^{5} + 3 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.f_d_au_ak |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abi_a_bhj |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 21 T^{2} + 24 T^{3} + 368 T^{4} + 24 p T^{5} - 21 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ad_av_y_oe |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.23.ai_em_awm_gje |
| 29 | $D_{4}$ | \( ( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.o_gn_bvy_mam |
| 37 | $D_4\times C_2$ | \( 1 + 9 T - 3 T^{2} + 90 T^{3} + 2690 T^{4} + 90 p T^{5} - 3 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.j_ad_dm_dzm |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 14 T^{2} + 192 T^{3} - 657 T^{4} + 192 p T^{5} - 14 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ag_ao_hk_azh |
| 43 | $D_4\times C_2$ | \( 1 - 3 T - 69 T^{2} + 24 T^{3} + 3848 T^{4} + 24 p T^{5} - 69 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ad_acr_y_fsa |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.e_ei_po_lko |
| 53 | $D_4\times C_2$ | \( 1 + 3 T - 89 T^{2} - 24 T^{3} + 6318 T^{4} - 24 p T^{5} - 89 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.d_adl_ay_jja |
| 59 | $D_4\times C_2$ | \( 1 + 11 T - 17 T^{2} + 220 T^{3} + 8424 T^{4} + 220 p T^{5} - 17 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.l_ar_im_mma |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.61.ai_ki_acfo_bljy |
| 67 | $D_4\times C_2$ | \( 1 + 2 T - 90 T^{2} - 80 T^{3} + 4079 T^{4} - 80 p T^{5} - 90 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.c_adm_adc_gax |
| 71 | $D_4\times C_2$ | \( 1 + 2 T - 98 T^{2} - 80 T^{3} + 5079 T^{4} - 80 p T^{5} - 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.c_adu_adc_hnj |
| 73 | $D_4\times C_2$ | \( 1 - 25 T + 333 T^{2} - 50 p T^{3} + 470 p T^{4} - 50 p^{2} T^{5} + 333 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.az_mv_afkk_bytq |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) | 4.79.ai_aeg_aey_bcbz |
| 83 | $D_4\times C_2$ | \( 1 + 7 T - 119 T^{2} + 14 T^{3} + 17268 T^{4} + 14 p T^{5} - 119 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.h_aep_o_zoe |
| 89 | $D_{4}$ | \( ( 1 - 14 T + 186 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.abc_vw_alke_ewfu |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 189 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.y_uc_kea_esad |
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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
−6.87635805813206890984050844081, −6.53619457240208209292965739448, −6.29534340879504744853064795102, −6.12350000475510081251891722861, −5.93269013567979451754999543251, −5.83954203973516253512386815768, −5.34411157041253052774355778926, −5.28085283972847041638061849290, −4.95926618319653553578218120911, −4.89731270042966825232194502123, −4.75892980464647616574975499459, −4.22514131493890083554723156797, −3.97412669208327647343006078219, −3.60531030072974941994941944024, −3.44061858647686472375250871467, −3.36497477102111830009675469461, −3.08420563733401085567676544107, −2.69541589093673799724234577458, −2.48332435743456107065219893378, −2.31118434414065862991905898529, −1.85069693516036487357659790873, −1.84709562975736769161361549231, −1.33729307844830083241256401815, −0.899865752698742268869259439984, −0.43197982683158562984954360631,
0.43197982683158562984954360631, 0.899865752698742268869259439984, 1.33729307844830083241256401815, 1.84709562975736769161361549231, 1.85069693516036487357659790873, 2.31118434414065862991905898529, 2.48332435743456107065219893378, 2.69541589093673799724234577458, 3.08420563733401085567676544107, 3.36497477102111830009675469461, 3.44061858647686472375250871467, 3.60531030072974941994941944024, 3.97412669208327647343006078219, 4.22514131493890083554723156797, 4.75892980464647616574975499459, 4.89731270042966825232194502123, 4.95926618319653553578218120911, 5.28085283972847041638061849290, 5.34411157041253052774355778926, 5.83954203973516253512386815768, 5.93269013567979451754999543251, 6.12350000475510081251891722861, 6.29534340879504744853064795102, 6.53619457240208209292965739448, 6.87635805813206890984050844081
