| L(s) = 1 | − 2·3-s + 9·5-s + 9-s + 6·11-s − 15·13-s − 18·15-s − 6·17-s + 3·19-s + 39·25-s + 2·27-s − 12·33-s + 6·37-s + 30·39-s − 13·41-s + 9·45-s − 16·47-s − 7·49-s + 12·51-s + 16·53-s + 54·55-s − 6·57-s + 21·59-s − 12·61-s − 135·65-s − 67-s − 4·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4.02·5-s + 1/3·9-s + 1.80·11-s − 4.16·13-s − 4.64·15-s − 1.45·17-s + 0.688·19-s + 39/5·25-s + 0.384·27-s − 2.08·33-s + 0.986·37-s + 4.80·39-s − 2.03·41-s + 1.34·45-s − 2.33·47-s − 49-s + 1.68·51-s + 2.19·53-s + 7.28·55-s − 0.794·57-s + 2.73·59-s − 1.53·61-s − 16.7·65-s − 0.122·67-s − 0.474·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360870685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360870685\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 9 T + 42 T^{2} - 27 p T^{3} + 67 p T^{4} - 27 p^{2} T^{5} + 42 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.aj_bq_aff_mx |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) | 4.7.a_h_a_a |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ag_bv_agy_bev |
| 13 | $D_4\times C_2$ | \( 1 + 15 T + 118 T^{2} + 645 T^{3} + 2655 T^{4} + 645 p T^{5} + 118 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.p_eo_yv_dyd |
| 17 | $C_2^2$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.g_bx_io_bxk |
| 19 | $D_4\times C_2$ | \( 1 - 3 T + 40 T^{2} - 111 T^{3} + 1065 T^{4} - 111 p T^{5} + 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ad_bo_aeh_boz |
| 23 | $D_4\times C_2$ | \( 1 - 3 T^{2} + 929 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ad_a_bjt |
| 29 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aeg_a_gzb |
| 31 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1950 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ace_a_cxa |
| 41 | $D_4\times C_2$ | \( 1 + 13 T + 50 T^{2} + 481 T^{3} + 5551 T^{4} + 481 p T^{5} + 50 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.n_by_sn_ifn |
| 43 | $D_4\times C_2$ | \( 1 - 167 T^{2} + 10665 T^{4} - 167 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_agl_a_puf |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 89 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.q_ji_dfs_bbdz |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 107 T^{2} - 688 T^{3} + 5824 T^{4} - 688 p T^{5} + 107 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.aq_ed_abam_iqa |
| 59 | $D_4\times C_2$ | \( 1 - 21 T + 286 T^{2} - 2919 T^{3} + 24513 T^{4} - 2919 p T^{5} + 286 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.av_la_aeih_bkgv |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 10443 T^{4} + 1272 p T^{5} + 154 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.m_fy_bwy_plr |
| 67 | $D_4\times C_2$ | \( 1 + T - 128 T^{2} - 5 T^{3} + 12085 T^{4} - 5 p T^{5} - 128 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.67.b_aey_af_rwv |
| 71 | $D_4\times C_2$ | \( 1 + 4 T - 46 T^{2} - 320 T^{3} - 2333 T^{4} - 320 p T^{5} - 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.e_abu_ami_adlt |
| 73 | $D_{4}$ | \( ( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.e_adc_eu_tgg |
| 79 | $D_4\times C_2$ | \( 1 + 9 T + 148 T^{2} + 1089 T^{3} + 10533 T^{4} + 1089 p T^{5} + 148 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.j_fs_bpx_ppd |
| 83 | $D_4\times C_2$ | \( 1 - 14 T + 2 T^{2} - 392 T^{3} + 14479 T^{4} - 392 p T^{5} + 2 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ao_c_apc_vkx |
| 89 | $D_4\times C_2$ | \( 1 - 30 T + 490 T^{2} - 5700 T^{3} + 54879 T^{4} - 5700 p T^{5} + 490 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.abe_sw_ailg_ddet |
| 97 | $D_4\times C_2$ | \( 1 - 227 T^{2} + 25269 T^{4} - 227 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ait_a_bljx |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86764278746090935493762137854, −7.79904924400233327466142179003, −7.29818754290870853533673624458, −7.12535543519158306061131845852, −6.96367306037660345376970548719, −6.62693296630144699850604104165, −6.49333252062224199800646906147, −6.28110116116793554076849117725, −6.11942716326835370032412221890, −5.80242888948068270648543950730, −5.44420901701393670502133083922, −5.35567725405135821361300909978, −4.97068385266799375650502006116, −4.94521613227002102642862566277, −4.67157970288311480383182166492, −4.45712252383345080275487200664, −3.88102656448249022878361636369, −3.32120507966397985979596947426, −3.06429220976426916792342612684, −2.33283869257859256342312621777, −2.30001298623258855455797941412, −2.29632598435277872632752667342, −1.66124289423111128507702649091, −1.50406318103013110175424909662, −0.54128237400190053152063453537,
0.54128237400190053152063453537, 1.50406318103013110175424909662, 1.66124289423111128507702649091, 2.29632598435277872632752667342, 2.30001298623258855455797941412, 2.33283869257859256342312621777, 3.06429220976426916792342612684, 3.32120507966397985979596947426, 3.88102656448249022878361636369, 4.45712252383345080275487200664, 4.67157970288311480383182166492, 4.94521613227002102642862566277, 4.97068385266799375650502006116, 5.35567725405135821361300909978, 5.44420901701393670502133083922, 5.80242888948068270648543950730, 6.11942716326835370032412221890, 6.28110116116793554076849117725, 6.49333252062224199800646906147, 6.62693296630144699850604104165, 6.96367306037660345376970548719, 7.12535543519158306061131845852, 7.29818754290870853533673624458, 7.79904924400233327466142179003, 7.86764278746090935493762137854
