| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s + 12·7-s + 8·8-s + 16·10-s + 2·11-s + 8·13-s + 48·14-s − 4·16-s − 16·17-s − 12·19-s + 32·20-s + 8·22-s + 12·23-s + 11·25-s + 32·26-s + 96·28-s − 8·31-s − 32·32-s − 64·34-s + 48·35-s − 24·37-s − 48·38-s + 32·40-s + 18·41-s + 14·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 1.78·5-s + 4.53·7-s + 2.82·8-s + 5.05·10-s + 0.603·11-s + 2.21·13-s + 12.8·14-s − 16-s − 3.88·17-s − 2.75·19-s + 7.15·20-s + 1.70·22-s + 2.50·23-s + 11/5·25-s + 6.27·26-s + 18.1·28-s − 1.43·31-s − 5.65·32-s − 10.9·34-s + 8.11·35-s − 3.94·37-s − 7.78·38-s + 5.05·40-s + 2.81·41-s + 2.13·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(26.46339426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.46339426\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + p T^{2} + 8 T^{3} - 44 T^{4} + 8 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ae_f_i_abs |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 300 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.am_cv_alo_bjc |
| 11 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 26 T^{3} - 32 T^{4} - 26 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_f_aba_abg |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) | 4.13.ai_r_ee_abbw |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.17.q_gi_bpg_huc |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.m_cu_rc_dow |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.am_dt_awq_ekm |
| 29 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 156 T^{3} - 484 T^{4} + 156 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_j_ga_asq |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 88 p T^{5} + 13 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.i_n_adk_ang |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.y_lc_drc_yyo |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 1314 p T^{5} + 181 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.as_gz_abyo_lyq |
| 43 | $D_4\times C_2$ | \( 1 - 14 T + 65 T^{2} + 474 T^{3} - 6280 T^{4} + 474 p T^{5} + 65 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ao_cn_sg_ajho |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 19 T^{2} - 88 T^{3} + 1672 T^{4} - 88 p T^{5} - 19 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.i_at_adk_cmi |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 11806 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.q_ey_bwq_rmc |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 462 T^{3} + 848 T^{4} - 462 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ag_bt_aru_bgq |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 2088 p T^{5} + 261 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.m_kb_dci_bkem |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 762 T^{3} + 2600 T^{4} + 762 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.c_cn_bdi_dwa |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_aga_a_tjq |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_ajc_a_bjfy |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_afz_a_baia |
| 83 | $D_4\times C_2$ | \( 1 - 22 T + 185 T^{2} + 290 T^{3} - 14024 T^{4} + 290 p T^{5} + 185 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.aw_hd_le_autk |
| 89 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_bk_a_tjq |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.c_ahj_c_bpts |
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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
−8.346760097979251694027590529866, −7.65074202273372157566079741528, −7.52225300487081099952851990246, −7.04985606924853834159894697493, −6.94361858775481856162498027275, −6.51473834886394165267783479073, −6.50152729609205577339568083303, −6.10506541828453617118233986961, −6.07125515995286220928475998199, −5.71325454549113683281076716226, −5.38765840890199586588349505677, −5.02471312083111469700872889468, −4.84857110397660473702567812082, −4.75306077083301007876810933407, −4.64447216272210668371467189189, −4.36342768056604405552351597458, −4.05309217794302992536761349981, −3.71251742116378939647714884110, −3.50438069768030824563392271604, −2.73817215348547152477752612920, −2.44731318008402353571952307823, −2.06062063676510350442717570419, −1.85623255560960312270627319693, −1.77942955101796915401759718895, −1.28215177371447710200911774828,
1.28215177371447710200911774828, 1.77942955101796915401759718895, 1.85623255560960312270627319693, 2.06062063676510350442717570419, 2.44731318008402353571952307823, 2.73817215348547152477752612920, 3.50438069768030824563392271604, 3.71251742116378939647714884110, 4.05309217794302992536761349981, 4.36342768056604405552351597458, 4.64447216272210668371467189189, 4.75306077083301007876810933407, 4.84857110397660473702567812082, 5.02471312083111469700872889468, 5.38765840890199586588349505677, 5.71325454549113683281076716226, 6.07125515995286220928475998199, 6.10506541828453617118233986961, 6.50152729609205577339568083303, 6.51473834886394165267783479073, 6.94361858775481856162498027275, 7.04985606924853834159894697493, 7.52225300487081099952851990246, 7.65074202273372157566079741528, 8.346760097979251694027590529866
