| L(s) = 1 | − 4·3-s + 2·5-s + 2·7-s + 6·9-s − 2·11-s − 2·13-s − 8·15-s − 16·19-s − 8·21-s − 6·23-s + 11·25-s + 4·27-s − 10·29-s + 10·31-s + 8·33-s + 4·35-s + 16·37-s + 8·39-s + 14·41-s + 10·43-s + 12·45-s + 2·47-s + 9·49-s + 16·53-s − 4·55-s + 64·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 0.755·7-s + 2·9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s − 3.67·19-s − 1.74·21-s − 1.25·23-s + 11/5·25-s + 0.769·27-s − 1.85·29-s + 1.79·31-s + 1.39·33-s + 0.676·35-s + 2.63·37-s + 1.28·39-s + 2.18·41-s + 1.52·43-s + 1.78·45-s + 0.291·47-s + 9/7·49-s + 2.19·53-s − 0.539·55-s + 8.47·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8362492635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8362492635\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ac_ah_ac_cy |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ac_af_k_e |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 10 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_an_ak_eu |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 46 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.c_b_abu_aie |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_u_a_bac |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.19.q_gq_bsy_izm |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 18 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.g_an_s_boe |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 10 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.k_bp_k_aka |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 190 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ak_t_ahi_dse |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.aq_ho_aclk_rna |
| 41 | $D_4\times C_2$ | \( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 350 p T^{5} + 89 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ao_dl_anm_coq |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) | 4.43.ak_br_le_aeho |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 106 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ac_abl_ec_abbo |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.aq_ka_adoy_bgok |
| 59 | $D_4\times C_2$ | \( 1 - 14 T + 83 T^{2} + 70 T^{3} - 2276 T^{4} + 70 p T^{5} + 83 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ao_df_cs_adjo |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 90 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ag_act_dm_ieu |
| 67 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 290 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ak_af_le_agi |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ai_em_ablg_vzi |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_jk_a_bluk |
| 79 | $D_4\times C_2$ | \( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 2530 p T^{5} + 211 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aw_id_adti_bslw |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 18 p T^{5} - 133 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.g_afd_s_bbqq |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bg_baq_onw_gfdu |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.c_agl_abu_bctw |
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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
−8.528494844718235248589158491172, −8.356799357655653873601405921314, −8.277808320956307344423857159326, −7.69232019076556994287757704351, −7.66031989695406549814228213055, −7.05741419137778495630014040907, −6.94507973022445647786709110214, −6.68910171845857904948459346737, −6.30738659467173641567699437648, −6.24416714420139548338665191609, −5.73387370260748724227880405511, −5.73263954580508246532527056950, −5.67762622913793743669812234480, −5.06435268804208249331100137762, −4.99058025842595815179822458880, −4.48761440104941814252176211708, −4.30408810415546339241063490909, −4.01275612175959312449128796544, −3.94352999670295146950212585372, −2.72527788643027124441115982017, −2.52742618224521092967301665505, −2.39291616219887072688479580107, −2.02961774183538419465065363840, −0.929514809948609905241792156450, −0.66540860896531064584168366888,
0.66540860896531064584168366888, 0.929514809948609905241792156450, 2.02961774183538419465065363840, 2.39291616219887072688479580107, 2.52742618224521092967301665505, 2.72527788643027124441115982017, 3.94352999670295146950212585372, 4.01275612175959312449128796544, 4.30408810415546339241063490909, 4.48761440104941814252176211708, 4.99058025842595815179822458880, 5.06435268804208249331100137762, 5.67762622913793743669812234480, 5.73263954580508246532527056950, 5.73387370260748724227880405511, 6.24416714420139548338665191609, 6.30738659467173641567699437648, 6.68910171845857904948459346737, 6.94507973022445647786709110214, 7.05741419137778495630014040907, 7.66031989695406549814228213055, 7.69232019076556994287757704351, 8.277808320956307344423857159326, 8.356799357655653873601405921314, 8.528494844718235248589158491172
