| L(s) = 1 | + 2·3-s + 5-s − 7-s − 9-s + 7·11-s + 10·13-s + 2·15-s + 5·17-s + 4·19-s − 2·21-s − 8·23-s − 25-s − 2·27-s + 10·29-s − 6·31-s + 14·33-s − 35-s + 14·37-s + 20·39-s − 2·41-s + 3·43-s − 45-s − 3·47-s − 10·49-s + 10·51-s + 4·53-s + 7·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s − 1/3·9-s + 2.11·11-s + 2.77·13-s + 0.516·15-s + 1.21·17-s + 0.917·19-s − 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.384·27-s + 1.85·29-s − 1.07·31-s + 2.43·33-s − 0.169·35-s + 2.30·37-s + 3.20·39-s − 0.312·41-s + 0.457·43-s − 0.149·45-s − 0.437·47-s − 1.42·49-s + 1.40·51-s + 0.549·53-s + 0.943·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.922920395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.922920395\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ac_f_ak_bc |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 2 T^{2} + 9 T^{3} + 2 T^{4} + 9 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ab_c_j_c |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 11 T^{2} + 40 T^{3} + 60 T^{4} + 40 p T^{5} + 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.7.b_l_bo_ci |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 4 p T^{2} - 203 T^{3} + 742 T^{4} - 203 p T^{5} + 4 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ah_bs_ahv_bco |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 59 T^{2} - 232 T^{3} + 852 T^{4} - 232 p T^{5} + 59 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ak_ch_aiy_bgu |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + p T^{2} - 114 T^{3} + 698 T^{4} - 114 p T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.af_r_aek_baw |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 75 T^{2} + 420 T^{3} + 2456 T^{4} + 420 p T^{5} + 75 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.i_cx_qe_dqm |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 123 T^{2} - 712 T^{3} + 5108 T^{4} - 712 p T^{5} + 123 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ak_et_abbk_hom |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 68 T^{2} + 142 T^{3} + 1782 T^{4} + 142 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.g_cq_fm_cqo |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 152 T^{2} - 1018 T^{3} + 7006 T^{4} - 1018 p T^{5} + 152 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ao_fw_abne_kjm |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 92 T^{2} + 54 T^{3} + 4694 T^{4} + 54 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.c_do_cc_gyo |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 140 T^{2} - 291 T^{3} + 8278 T^{4} - 291 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ad_fk_alf_mgk |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 80 T^{2} + 247 T^{3} + 3038 T^{4} + 247 p T^{5} + 80 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.d_dc_jn_emw |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 143 T^{2} - 252 T^{3} + 9032 T^{4} - 252 p T^{5} + 143 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ae_fn_ajs_njk |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 321 T^{2} - 3390 T^{3} + 30764 T^{4} - 3390 p T^{5} + 321 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.au_mj_afak_btng |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 178 T^{2} + 521 T^{3} + 14618 T^{4} + 521 p T^{5} + 178 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.d_gw_ub_vqg |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 193 T^{2} - 1006 T^{3} + 16060 T^{4} - 1006 p T^{5} + 193 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ai_hl_abms_xts |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 30 T + 552 T^{2} + 6814 T^{3} + 65870 T^{4} + 6814 p T^{5} + 552 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.be_vg_kcc_dtlm |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 245 T^{2} - 1662 T^{3} + 25114 T^{4} - 1662 p T^{5} + 245 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aj_jl_acly_bldy |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 216 T^{2} - 1082 T^{3} + 18510 T^{4} - 1082 p T^{5} + 216 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ak_ii_abpq_bbjy |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 216 T^{2} - 212 T^{3} + 21054 T^{4} - 212 p T^{5} + 216 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ae_ii_aie_bfdu |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 152 T^{2} + 224 T^{3} - 4082 T^{4} + 224 p T^{5} + 152 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.q_fw_iq_agba |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 328 T^{2} + 1658 T^{3} + 45422 T^{4} + 1658 p T^{5} + 328 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.g_mq_clu_cpfa |
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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
−7.911144679977246051685050971378, −7.42438102823128877433861103980, −7.32449518994707253871917663539, −6.77380938459330309668909540662, −6.67379065150857517696529135388, −6.51565818587038843587150487508, −6.14868938462097966152615195952, −6.07776914851798056103268562789, −5.95782901618276610151341240399, −5.42583048296308275391962889041, −5.41551883122850368212255801765, −5.14516018129726910240316557744, −4.43156516844479770717600601933, −4.31576405370537100327769904158, −3.95856135211899794599950600395, −3.80460682512129720160823924460, −3.71682930540673225382122132388, −3.15003659553952519023511169874, −3.12646916284193874369756394439, −2.66945377331035666815973648793, −2.51245460756517635952807766016, −1.81046502848137566519298407966, −1.40446750846905959674054433858, −1.28544118175343581401372569750, −0.805353815246628128701971115012,
0.805353815246628128701971115012, 1.28544118175343581401372569750, 1.40446750846905959674054433858, 1.81046502848137566519298407966, 2.51245460756517635952807766016, 2.66945377331035666815973648793, 3.12646916284193874369756394439, 3.15003659553952519023511169874, 3.71682930540673225382122132388, 3.80460682512129720160823924460, 3.95856135211899794599950600395, 4.31576405370537100327769904158, 4.43156516844479770717600601933, 5.14516018129726910240316557744, 5.41551883122850368212255801765, 5.42583048296308275391962889041, 5.95782901618276610151341240399, 6.07776914851798056103268562789, 6.14868938462097966152615195952, 6.51565818587038843587150487508, 6.67379065150857517696529135388, 6.77380938459330309668909540662, 7.32449518994707253871917663539, 7.42438102823128877433861103980, 7.911144679977246051685050971378
