| L(s) = 1 | − 6·5-s + 2·7-s + 9·11-s + 4·17-s − 11·19-s + 4·23-s + 9·25-s + 3·29-s − 16·31-s − 12·35-s − 4·37-s − 10·41-s + 43-s − 32·47-s + 49-s − 54·55-s − 4·59-s − 26·67-s + 2·71-s + 16·73-s + 18·77-s + 32·79-s + 32·83-s − 24·85-s + 21·89-s + 66·95-s + 23·97-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 0.755·7-s + 2.71·11-s + 0.970·17-s − 2.52·19-s + 0.834·23-s + 9/5·25-s + 0.557·29-s − 2.87·31-s − 2.02·35-s − 0.657·37-s − 1.56·41-s + 0.152·43-s − 4.66·47-s + 1/7·49-s − 7.28·55-s − 0.520·59-s − 3.17·67-s + 0.237·71-s + 1.87·73-s + 2.05·77-s + 3.60·79-s + 3.51·83-s − 2.60·85-s + 2.22·89-s + 6.77·95-s + 2.33·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.691222456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691222456\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | |
| good | 5 | $D_{4}$ | \( ( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.g_bb_dg_in |
| 11 | $D_4\times C_2$ | \( 1 - 9 T + 42 T^{2} - 153 T^{3} + 509 T^{4} - 153 p T^{5} + 42 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.aj_bq_afx_tp |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 9 T^{2} + 36 T^{3} + 64 T^{4} + 36 p T^{5} - 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_aj_bk_cm |
| 19 | $D_4\times C_2$ | \( 1 + 11 T + 56 T^{2} + 297 T^{3} + 1565 T^{4} + 297 p T^{5} + 56 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.l_ce_ll_cif |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_abi_aq_cjr |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 48 T^{2} + 3 T^{3} + 2147 T^{4} + 3 p T^{5} - 48 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ad_abw_d_dep |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 65 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.q_hm_chc_ozb |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 192 T^{3} - 1285 T^{4} - 192 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_ak_ahk_abxl |
| 41 | $D_4\times C_2$ | \( 1 + 10 T + 6 T^{2} + 120 T^{3} + 3055 T^{4} + 120 p T^{5} + 6 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.k_g_eq_enn |
| 43 | $D_4\times C_2$ | \( 1 - T - 56 T^{2} + 29 T^{3} + 1357 T^{4} + 29 p T^{5} - 56 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ab_ace_bd_caf |
| 47 | $D_{4}$ | \( ( 1 + 16 T + 145 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.bg_va_jci_cvgd |
| 53 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aw_a_imt |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 93 T^{2} - 36 T^{3} + 7456 T^{4} - 36 p T^{5} - 93 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.e_adp_abk_lau |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} ) \) | 4.61.a_acs_a_btj |
| 67 | $C_2^2$ | \( ( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ba_oj_gna_ckem |
| 71 | $D_4\times C_2$ | \( 1 - 2 T - 22 T^{2} + 232 T^{3} - 4649 T^{4} + 232 p T^{5} - 22 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ac_aw_iy_agwv |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.aq_ky_aeiq_bvms |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.79.abg_bay_aogm_fvfi |
| 83 | $D_{4}$ | \( ( 1 - 16 T + 217 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.abg_bao_aofg_fwxn |
| 89 | $D_4\times C_2$ | \( 1 - 21 T + 182 T^{2} - 1701 T^{3} + 19911 T^{4} - 1701 p T^{5} + 182 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.av_ha_acnl_bdlv |
| 97 | $D_4\times C_2$ | \( 1 - 23 T + 232 T^{2} - 2369 T^{3} + 27487 T^{4} - 2369 p T^{5} + 232 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ax_iy_adnd_borf |
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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
−6.23770718570575231751660741211, −6.02371116051752236478948539343, −5.78095826371523633982622055700, −5.35479007642785097540434015242, −5.30705516189232694001702193981, −4.94750213286174060798523845880, −4.69540533863023991353702332443, −4.63825796533942156082904125432, −4.60208314492806176987508226773, −4.26549008394152380392483488658, −3.98056839962219945372495894175, −3.66447954601959377231006724289, −3.63604545443222128377956930471, −3.42880059121593833822951847182, −3.42879930713444514918568349927, −3.34522081437171042315846199206, −2.84756910355372802559312557680, −2.19299433963183152909420891690, −2.06772197225283396708146817354, −1.77982750376188790550898038105, −1.64859298955837659574575718634, −1.56714777379073054368861487598, −0.829301847491793867054176060408, −0.52752049676729223023238987346, −0.31328354565984077659220571801,
0.31328354565984077659220571801, 0.52752049676729223023238987346, 0.829301847491793867054176060408, 1.56714777379073054368861487598, 1.64859298955837659574575718634, 1.77982750376188790550898038105, 2.06772197225283396708146817354, 2.19299433963183152909420891690, 2.84756910355372802559312557680, 3.34522081437171042315846199206, 3.42879930713444514918568349927, 3.42880059121593833822951847182, 3.63604545443222128377956930471, 3.66447954601959377231006724289, 3.98056839962219945372495894175, 4.26549008394152380392483488658, 4.60208314492806176987508226773, 4.63825796533942156082904125432, 4.69540533863023991353702332443, 4.94750213286174060798523845880, 5.30705516189232694001702193981, 5.35479007642785097540434015242, 5.78095826371523633982622055700, 6.02371116051752236478948539343, 6.23770718570575231751660741211
