| L(s) = 1 | − 2·3-s + 2·7-s + 5·9-s − 2·11-s − 4·13-s + 4·17-s + 8·19-s − 4·21-s − 8·23-s + 8·25-s − 10·27-s + 4·29-s + 8·31-s + 4·33-s + 4·37-s + 8·39-s + 8·41-s − 32·43-s − 4·47-s + 7·49-s − 8·51-s − 8·53-s − 16·57-s − 6·59-s + 22·61-s + 10·63-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 5/3·9-s − 0.603·11-s − 1.10·13-s + 0.970·17-s + 1.83·19-s − 0.872·21-s − 1.66·23-s + 8/5·25-s − 1.92·27-s + 0.742·29-s + 1.43·31-s + 0.696·33-s + 0.657·37-s + 1.28·39-s + 1.24·41-s − 4.87·43-s − 0.583·47-s + 49-s − 1.12·51-s − 1.09·53-s − 2.11·57-s − 0.781·59-s + 2.81·61-s + 1.25·63-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.241228509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241228509\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.c_ab_ac_e |
| 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ai_a_bn |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.e_bq_ey_bex |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ae_aw_aq_bhz |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 12 T^{2} - 112 T^{3} + 1127 T^{4} - 112 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_m_aei_brj |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 4 T^{2} + 112 T^{3} + 1599 T^{4} + 112 p T^{5} + 4 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.i_e_ei_cjn |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.29.ae_es_ano_hzn |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 128 T^{3} - 829 T^{4} + 128 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ai_s_ey_abfx |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 12 T^{2} + 184 T^{3} - 1177 T^{4} + 184 p T^{5} - 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_am_hc_abth |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ai_fw_abho_ntq |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.43.bg_vk_jdo_ctik |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 74 T^{2} - 16 T^{3} + 5139 T^{4} - 16 p T^{5} - 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.e_acw_aq_hpr |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 56 T^{2} + 112 T^{3} + 8199 T^{4} + 112 p T^{5} - 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.i_ace_ei_mdj |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 89 T^{2} + 42 T^{3} + 10020 T^{4} + 42 p T^{5} - 89 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.g_adl_bq_ovk |
| 61 | $D_4\times C_2$ | \( 1 - 22 T + 249 T^{2} - 2486 T^{3} + 21980 T^{4} - 2486 p T^{5} + 249 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aw_jp_adrq_bgnk |
| 67 | $D_4\times C_2$ | \( 1 + 2 T - 81 T^{2} - 98 T^{3} + 2468 T^{4} - 98 p T^{5} - 81 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.c_add_adu_dqy |
| 71 | $D_{4}$ | \( ( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bo_bgq_rlw_greg |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 132 T^{2} - 8 T^{3} + 15407 T^{4} - 8 p T^{5} - 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ae_afc_ai_wup |
| 79 | $D_4\times C_2$ | \( 1 + 2 T - 105 T^{2} - 98 T^{3} + 5324 T^{4} - 98 p T^{5} - 105 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.c_aeb_adu_hwu |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.i_ie_cds_bmnm |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 262 T^{2} - 3264 T^{3} + 39411 T^{4} - 3264 p T^{5} + 262 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ay_kc_aevo_cghv |
| 97 | $D_{4}$ | \( ( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ca_cbi_biua_psgx |
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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
−7.47032624464068063795125073115, −7.23268746270647485100722028442, −7.22525180843555166633406375879, −7.20715712399365166480265638220, −6.72762034231786950890023965753, −6.39182043058938636872172565420, −6.22134086282649029077306141139, −5.97590244578886457962677390756, −5.65945098163931676913637486432, −5.52849839273620664665972224100, −5.11110098458588594968917133374, −4.80843712905335311978577114344, −4.80176709734628478224511175332, −4.78094598770218405033849484377, −4.26375492630593872601243152019, −3.99514800990315884886387213557, −3.54859808829563826622730859726, −3.22620695654665631689476086268, −3.06195983504856918908553014603, −2.50922963941114612604574900415, −2.43383566546140164621026429816, −1.65197379828808240891971839751, −1.38665609045316762860391163589, −1.20843327737567235018179603868, −0.38328396931174052839638251591,
0.38328396931174052839638251591, 1.20843327737567235018179603868, 1.38665609045316762860391163589, 1.65197379828808240891971839751, 2.43383566546140164621026429816, 2.50922963941114612604574900415, 3.06195983504856918908553014603, 3.22620695654665631689476086268, 3.54859808829563826622730859726, 3.99514800990315884886387213557, 4.26375492630593872601243152019, 4.78094598770218405033849484377, 4.80176709734628478224511175332, 4.80843712905335311978577114344, 5.11110098458588594968917133374, 5.52849839273620664665972224100, 5.65945098163931676913637486432, 5.97590244578886457962677390756, 6.22134086282649029077306141139, 6.39182043058938636872172565420, 6.72762034231786950890023965753, 7.20715712399365166480265638220, 7.22525180843555166633406375879, 7.23268746270647485100722028442, 7.47032624464068063795125073115
