| L(s) = 1 | + 2·3-s − 9-s + 6·13-s − 6·17-s + 4·23-s + 11·25-s − 2·27-s + 4·29-s + 12·39-s − 30·43-s + 7·49-s − 12·51-s − 16·53-s − 28·61-s + 8·69-s + 22·75-s − 28·79-s − 7·81-s + 8·87-s − 4·101-s − 4·103-s + 16·107-s + 40·113-s − 6·117-s + 8·121-s + 127-s − 60·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s + 1.66·13-s − 1.45·17-s + 0.834·23-s + 11/5·25-s − 0.384·27-s + 0.742·29-s + 1.92·39-s − 4.57·43-s + 49-s − 1.68·51-s − 2.19·53-s − 3.58·61-s + 0.963·69-s + 2.54·75-s − 3.15·79-s − 7/9·81-s + 0.857·87-s − 0.398·101-s − 0.394·103-s + 1.54·107-s + 3.76·113-s − 0.554·117-s + 8/11·121-s + 0.0887·127-s − 5.28·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056457089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056457089\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ac_f_ak_bc |
| 5 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 76 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_al_a_cy |
| 7 | $D_4\times C_2$ | \( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.7.a_ah_a_e |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ai_a_hi |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.g_cv_li_cvk |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_cm_aie_dek |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ae_dk_aky_flm |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adk_a_fpu |
| 37 | $D_4\times C_2$ | \( 1 - 27 T^{2} + 1964 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abb_a_cxo |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.41.a_abk_a_flu |
| 43 | $D_{4}$ | \( ( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.be_th_iaw_ckgy |
| 47 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 7444 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_aep_a_lai |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.q_gq_cnw_wre |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abk_a_kug |
| 61 | $D_{4}$ | \( ( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bc_tk_ixo_ddmc |
| 67 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15742 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ahc_a_xhm |
| 71 | $D_4\times C_2$ | \( 1 - 207 T^{2} + 20756 T^{4} - 207 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ahz_a_besi |
| 73 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_bg_a_hzy |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.bc_we_lds_enrq |
| 83 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aho_a_bipi |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ami_a_cjfi |
| 97 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 32614 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ajk_a_bwgk |
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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
−7.12445530459479445950339876041, −6.98759551463722955451535610529, −6.98497581440047599509218056559, −6.84332891740274639139725491282, −6.38457789416031475465091404854, −6.12686021533129156741440806561, −6.08494221905356219751254855900, −5.77711374292470199531780004736, −5.62891074136644239085714101155, −4.89029849281853505266445312117, −4.88115242942521584698908832902, −4.70311909119154350117598224502, −4.64156598928606728533311191339, −4.34141063039297270672079728674, −3.65577065447974136583248301532, −3.48254416501899484461848521978, −3.42328050443828810166108565578, −3.03402280629500291444028681074, −2.93046297261402931772719844874, −2.57446409088990163390061741889, −2.31586522764407868272721444559, −1.64919533581280531835925963654, −1.35748278734403408194317010611, −1.35687249803467095238169893017, −0.31160451230607153917183171838,
0.31160451230607153917183171838, 1.35687249803467095238169893017, 1.35748278734403408194317010611, 1.64919533581280531835925963654, 2.31586522764407868272721444559, 2.57446409088990163390061741889, 2.93046297261402931772719844874, 3.03402280629500291444028681074, 3.42328050443828810166108565578, 3.48254416501899484461848521978, 3.65577065447974136583248301532, 4.34141063039297270672079728674, 4.64156598928606728533311191339, 4.70311909119154350117598224502, 4.88115242942521584698908832902, 4.89029849281853505266445312117, 5.62891074136644239085714101155, 5.77711374292470199531780004736, 6.08494221905356219751254855900, 6.12686021533129156741440806561, 6.38457789416031475465091404854, 6.84332891740274639139725491282, 6.98497581440047599509218056559, 6.98759551463722955451535610529, 7.12445530459479445950339876041
