| L(s) = 1 | + 3·2-s − 6·3-s + 5·4-s − 2·5-s − 18·6-s + 2·7-s + 6·8-s + 21·9-s − 6·10-s − 30·12-s + 4·13-s + 6·14-s + 12·15-s + 4·16-s + 6·17-s + 63·18-s + 4·19-s − 10·20-s − 12·21-s − 7·23-s − 36·24-s + 6·25-s + 12·26-s − 54·27-s + 10·28-s + 3·29-s + 36·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 3.46·3-s + 5/2·4-s − 0.894·5-s − 7.34·6-s + 0.755·7-s + 2.12·8-s + 7·9-s − 1.89·10-s − 8.66·12-s + 1.10·13-s + 1.60·14-s + 3.09·15-s + 16-s + 1.45·17-s + 14.8·18-s + 0.917·19-s − 2.23·20-s − 2.61·21-s − 1.45·23-s − 7.34·24-s + 6/5·25-s + 2.35·26-s − 10.3·27-s + 1.88·28-s + 0.557·29-s + 6.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384433800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384433800\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} ) \) | 4.2.ad_e_ad_d |
| 5 | $D_4\times C_2$ | \( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 8 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.c_ac_ai_aj |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.7.ac_al_ac_fs |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 64 T^{3} - 277 T^{4} + 64 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_g_cm_akr |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ag_ch_ajs_cgb |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.19.ae_de_aiy_dod |
| 23 | $D_4\times C_2$ | \( 1 + 7 T - 8 T^{2} + 77 T^{3} + 1593 T^{4} + 77 p T^{5} - 8 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.h_ai_cz_cjh |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 40 T^{2} + 27 T^{3} + 1299 T^{4} + 27 p T^{5} - 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ad_abo_bb_bxz |
| 31 | $D_4\times C_2$ | \( 1 - T - 60 T^{2} + T^{3} + 2729 T^{4} + p T^{5} - 60 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ab_aci_b_eaz |
| 37 | $D_{4}$ | \( ( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ac_dj_age_gxh |
| 41 | $D_4\times C_2$ | \( 1 - 14 T + 85 T^{2} - 406 T^{3} + 2604 T^{4} - 406 p T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ao_dh_apq_dwe |
| 43 | $D_4\times C_2$ | \( 1 - 3 T - 68 T^{2} + 27 T^{3} + 3693 T^{4} + 27 p T^{5} - 68 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ad_acq_bb_fmb |
| 47 | $D_4\times C_2$ | \( 1 + 6 T + 13 T^{2} - 426 T^{3} - 3372 T^{4} - 426 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.g_n_aqk_aezs |
| 53 | $D_{4}$ | \( ( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.g_gh_bea_smv |
| 59 | $D_4\times C_2$ | \( 1 + T - 116 T^{2} - T^{3} + 10149 T^{4} - p T^{5} - 116 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.59.b_aem_ab_paj |
| 61 | $D_4\times C_2$ | \( 1 + 15 T + 58 T^{2} + 675 T^{3} + 10443 T^{4} + 675 p T^{5} + 58 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.p_cg_zz_plr |
| 67 | $C_2^2$ | \( ( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_aba_qq_vvv |
| 71 | $D_{4}$ | \( ( 1 + 17 T + 183 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bi_zf_mue_evef |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.o_nf_eom_cgif |
| 79 | $D_4\times C_2$ | \( 1 + T - 56 T^{2} - 101 T^{3} - 3071 T^{4} - 101 p T^{5} - 56 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.79.b_ace_adx_aeod |
| 83 | $D_4\times C_2$ | \( 1 - 24 T + 271 T^{2} - 3336 T^{3} + 38328 T^{4} - 3336 p T^{5} + 271 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ay_kl_aeyi_cese |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bk_bga_sgy_hwew |
| 97 | $C_2^2$ | \( ( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.m_adi_qq_brrn |
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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
−7.07860158247513307102877641801, −6.54624475420649585599203832980, −6.28039183867822664303564740729, −6.12253562375239598742995717008, −6.04684075219928056307418700278, −6.02546164427297503423270059782, −5.72683938149015522933572440420, −5.42171449915404409397030906850, −5.27345593400663661819141472852, −5.11624614482333973598648285901, −4.63820210518501911781817162024, −4.57904730018091289486143509854, −4.50406961190000356100548546088, −4.19288314661847002056685153516, −3.99420447496141463087184370223, −3.90337503905429889394054504999, −3.36528087059602518746001488627, −2.99670177131227431509855428730, −2.91744149349461372363109171587, −2.57334144487244405896516243438, −1.76633978104089392124781771173, −1.47327014756688435995562343488, −1.33608728608983832056940154644, −0.908093192107786269905698506227, −0.36181492393634094364124425648,
0.36181492393634094364124425648, 0.908093192107786269905698506227, 1.33608728608983832056940154644, 1.47327014756688435995562343488, 1.76633978104089392124781771173, 2.57334144487244405896516243438, 2.91744149349461372363109171587, 2.99670177131227431509855428730, 3.36528087059602518746001488627, 3.90337503905429889394054504999, 3.99420447496141463087184370223, 4.19288314661847002056685153516, 4.50406961190000356100548546088, 4.57904730018091289486143509854, 4.63820210518501911781817162024, 5.11624614482333973598648285901, 5.27345593400663661819141472852, 5.42171449915404409397030906850, 5.72683938149015522933572440420, 6.02546164427297503423270059782, 6.04684075219928056307418700278, 6.12253562375239598742995717008, 6.28039183867822664303564740729, 6.54624475420649585599203832980, 7.07860158247513307102877641801
