| L(s) = 1 | + 2·2-s + 4-s + 4·5-s − 2·7-s − 2·8-s − 4·9-s + 8·10-s − 2·11-s − 4·14-s − 4·16-s + 4·17-s − 8·18-s − 10·19-s + 4·20-s − 4·22-s − 12·23-s + 10·25-s − 2·28-s − 8·29-s − 16·31-s − 2·32-s + 8·34-s − 8·35-s − 4·36-s + 6·37-s − 20·38-s − 8·40-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.78·5-s − 0.755·7-s − 0.707·8-s − 4/3·9-s + 2.52·10-s − 0.603·11-s − 1.06·14-s − 16-s + 0.970·17-s − 1.88·18-s − 2.29·19-s + 0.894·20-s − 0.852·22-s − 2.50·23-s + 2·25-s − 0.377·28-s − 1.48·29-s − 2.87·31-s − 0.353·32-s + 1.37·34-s − 1.35·35-s − 2/3·36-s + 0.986·37-s − 3.24·38-s − 1.26·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.514867786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514867786\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | | \( 1 \) | |
| good | 3 | $C_2^3$ | \( 1 + 4 T^{2} + 7 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_e_a_h |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.7.c_al_c_fs |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 9 T^{2} - 18 T^{3} + 4 T^{4} - 18 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_aj_as_e |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 24 T^{3} + 223 T^{4} + 24 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_am_y_ip |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 47 T^{2} + 150 T^{3} + 548 T^{4} + 150 p T^{5} + 47 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.k_bv_fu_vc |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.m_ck_qq_egx |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 30 T^{2} - 192 T^{3} - 1541 T^{4} - 192 p T^{5} + 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.i_be_ahk_achh |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.q_hs_ciy_pok |
| 37 | $D_4\times C_2$ | \( 1 - 6 T - p T^{2} + 6 T^{3} + 2628 T^{4} + 6 p T^{5} - p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ag_abl_g_dxc |
| 41 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 192 T^{3} - 941 T^{4} - 192 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.i_g_ahk_abkf |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.e_acw_q_ifz |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) | 4.47.am_ji_acrg_bbgd |
| 53 | $D_{4}$ | \( ( 1 - 2 T + 97 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.ae_hq_axc_wwh |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 30 T^{2} - 192 T^{3} + 1579 T^{4} - 192 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.i_abe_ahk_cit |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 20 T^{2} - 344 T^{3} - 3401 T^{4} - 344 p T^{5} - 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.e_au_ang_afav |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 46 T^{2} + 192 T^{3} + 3323 T^{4} + 192 p T^{5} - 46 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ai_abu_hk_exv |
| 71 | $D_4\times C_2$ | \( 1 - 4 T - 90 T^{2} + 144 T^{3} + 5059 T^{4} + 144 p T^{5} - 90 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ae_adm_fo_hmp |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.y_su_ise_dmqk |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.q_iy_dwi_brwk |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_fw_a_bcyc |
| 89 | $D_4\times C_2$ | \( 1 + 2 T - 135 T^{2} - 78 T^{3} + 11044 T^{4} - 78 p T^{5} - 135 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.c_aff_ada_qiu |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 92 T^{2} + 344 T^{3} + 703 T^{4} + 344 p T^{5} - 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ae_ado_ng_bbb |
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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
−6.58456270256597283268203176552, −6.20815825053605016885832455107, −5.90037375158691252010879185511, −5.81296760225233516228946147717, −5.75872992768782801136843030177, −5.73575470905809499385200293515, −5.59249293165674962702922677349, −5.41803628512423703861565240626, −4.76910375395677810030618244805, −4.68291725287655225308911761241, −4.60824836313431198272917003837, −4.23417198300907107918373203153, −4.01853317262002994531600476148, −3.70106913843537200144231823593, −3.45888750452691413518644949529, −3.34256982062449950076818960894, −3.18495571913824688451816181721, −2.64132335653936571888736078969, −2.34961572065047244596005419387, −2.32810798864093202647060255604, −2.05435492968560156900319530169, −1.73713979760129143437189153737, −1.49794592868965267540392876315, −0.60150111648538423660731139334, −0.21226731422369276871985792993,
0.21226731422369276871985792993, 0.60150111648538423660731139334, 1.49794592868965267540392876315, 1.73713979760129143437189153737, 2.05435492968560156900319530169, 2.32810798864093202647060255604, 2.34961572065047244596005419387, 2.64132335653936571888736078969, 3.18495571913824688451816181721, 3.34256982062449950076818960894, 3.45888750452691413518644949529, 3.70106913843537200144231823593, 4.01853317262002994531600476148, 4.23417198300907107918373203153, 4.60824836313431198272917003837, 4.68291725287655225308911761241, 4.76910375395677810030618244805, 5.41803628512423703861565240626, 5.59249293165674962702922677349, 5.73575470905809499385200293515, 5.75872992768782801136843030177, 5.81296760225233516228946147717, 5.90037375158691252010879185511, 6.20815825053605016885832455107, 6.58456270256597283268203176552
