| L(s) = 1 | + 6·5-s − 8·7-s − 4·11-s + 4·17-s + 8·19-s + 4·23-s + 19·25-s + 2·29-s + 12·31-s − 48·35-s − 6·37-s − 6·41-s + 4·43-s − 2·47-s + 34·49-s + 16·53-s − 24·55-s + 18·59-s − 12·61-s − 4·67-s + 6·71-s − 6·73-s + 32·77-s + 12·79-s − 28·83-s + 24·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 3.02·7-s − 1.20·11-s + 0.970·17-s + 1.83·19-s + 0.834·23-s + 19/5·25-s + 0.371·29-s + 2.15·31-s − 8.11·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 34/7·49-s + 2.19·53-s − 3.23·55-s + 2.34·59-s − 1.53·61-s − 0.488·67-s + 0.712·71-s − 0.702·73-s + 3.64·77-s + 1.35·79-s − 3.07·83-s + 2.60·85-s + 0.423·89-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
| \(L(1)\) |
\(\approx\) |
\(0.7176962984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7176962984\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ag_r_acc_ga |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.e_y_dg_qo |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.19.ai_dw_asu_epa |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 21 T^{2} + 36 T^{3} + 472 T^{4} + 36 p T^{5} - 21 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ae_av_bk_se |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 102 T^{3} - 908 T^{4} + 102 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ac_ad_dy_abiy |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 276 T^{3} + 1800 T^{4} - 276 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.am_ch_akq_crg |
| 37 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 258 T^{3} - 1740 T^{4} - 258 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.g_f_ajy_acoy |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.g_acd_cc_hpc |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 61 T^{2} + 36 T^{3} + 3392 T^{4} + 36 p T^{5} - 61 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_acj_bk_fam |
| 47 | $D_4\times C_2$ | \( 1 + 2 T - 39 T^{2} - 102 T^{3} - 548 T^{4} - 102 p T^{5} - 39 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.c_abn_ady_avc |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 99 T^{2} - 816 T^{3} + 8272 T^{4} - 816 p T^{5} + 99 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.aq_dv_abfk_mge |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.as_ev_acec_zea |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.m_km_deq_bmcs |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.e_bo_nc_ook |
| 71 | $C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ag_ael_acc_wmu |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 67 T^{2} - 258 T^{3} + 2652 T^{4} - 258 p T^{5} - 67 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.g_acp_ajy_dya |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 37 T^{2} - 276 T^{3} + 15144 T^{4} - 276 p T^{5} - 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.am_abl_akq_wkm |
| 83 | $D_{4}$ | \( ( 1 + 14 T + 202 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.bc_xc_luy_eyws |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 153 T^{2} + 36 T^{3} + 19216 T^{4} + 36 p T^{5} - 153 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ae_afx_bk_bclc |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 29 T^{2} + 808 T^{3} - 6968 T^{4} + 808 p T^{5} - 29 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_abd_bfc_akia |
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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
−6.03690192907921879435450295736, −5.92270993217392404188569791324, −5.74970637128859107220684149103, −5.63901433290271662882033062957, −5.42586248055096457160119856235, −5.20900469616996727047929121740, −5.08035871661537475829205361584, −4.69394769339451419388764193195, −4.68269685792458140903259557710, −4.39154725680430079599598646103, −3.85122376197854190836076040188, −3.74542202322772251876036517011, −3.43187292524303270495657240704, −3.23205401007979423873014784680, −3.21411188116625592028927876867, −2.95874704467622875122165543660, −2.53116993506486415703282903835, −2.41873997439199339824494777392, −2.31756486092501117727214018329, −2.31063050607431387676843782517, −1.42295946101361467895573275683, −1.26816065843818828775927677500, −1.11266705218093144008622675736, −0.831000436190284774277928330044, −0.11473523957196188851599905535,
0.11473523957196188851599905535, 0.831000436190284774277928330044, 1.11266705218093144008622675736, 1.26816065843818828775927677500, 1.42295946101361467895573275683, 2.31063050607431387676843782517, 2.31756486092501117727214018329, 2.41873997439199339824494777392, 2.53116993506486415703282903835, 2.95874704467622875122165543660, 3.21411188116625592028927876867, 3.23205401007979423873014784680, 3.43187292524303270495657240704, 3.74542202322772251876036517011, 3.85122376197854190836076040188, 4.39154725680430079599598646103, 4.68269685792458140903259557710, 4.69394769339451419388764193195, 5.08035871661537475829205361584, 5.20900469616996727047929121740, 5.42586248055096457160119856235, 5.63901433290271662882033062957, 5.74970637128859107220684149103, 5.92270993217392404188569791324, 6.03690192907921879435450295736
