| L(s) = 1 | + 2·3-s − 8·4-s + 2·5-s + 9-s + 11-s − 16·12-s − 13-s + 4·15-s + 40·16-s + 8·17-s + 7·19-s − 16·20-s + 8·23-s + 25-s − 2·27-s − 6·29-s − 9·31-s + 2·33-s − 8·36-s − 7·37-s − 2·39-s − 7·41-s − 3·43-s − 8·44-s + 2·45-s − 4·47-s + 80·48-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4·4-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 4.61·12-s − 0.277·13-s + 1.03·15-s + 10·16-s + 1.94·17-s + 1.60·19-s − 3.57·20-s + 1.66·23-s + 1/5·25-s − 0.384·27-s − 1.11·29-s − 1.61·31-s + 0.348·33-s − 4/3·36-s − 1.15·37-s − 0.320·39-s − 1.09·41-s − 0.457·43-s − 1.20·44-s + 0.298·45-s − 0.583·47-s + 11.5·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526594795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526594795\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| 31 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.2.a_i_a_y |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ao_a_fr |
| 11 | $D_4\times C_2$ | \( 1 - T + 3 T^{2} + 24 T^{3} - 128 T^{4} + 24 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ab_d_y_aey |
| 13 | $D_4\times C_2$ | \( 1 + T - T^{2} - 24 T^{3} - 178 T^{4} - 24 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.13.b_ab_ay_agw |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_o_aey_brf |
| 19 | $D_4\times C_2$ | \( 1 - 7 T + 23 T^{2} + 84 T^{3} - 616 T^{4} + 84 p T^{5} + 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ah_x_dg_axs |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.23.ai_em_awm_gje |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.g_dd_pa_feq |
| 37 | $D_4\times C_2$ | \( 1 + 7 T - 13 T^{2} - 84 T^{3} + 662 T^{4} - 84 p T^{5} - 13 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.h_an_adg_zm |
| 41 | $D_4\times C_2$ | \( 1 + 7 T - 21 T^{2} - 84 T^{3} + 1210 T^{4} - 84 p T^{5} - 21 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.h_av_adg_buo |
| 43 | $D_4\times C_2$ | \( 1 + 3 T - 55 T^{2} - 66 T^{3} + 1860 T^{4} - 66 p T^{5} - 55 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.d_acd_aco_cto |
| 47 | $D_{4}$ | \( ( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.e_a_gy_hco |
| 53 | $D_4\times C_2$ | \( 1 + 2 T - 6 T^{2} - 192 T^{3} - 2921 T^{4} - 192 p T^{5} - 6 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.c_ag_ahk_aeij |
| 59 | $D_4\times C_2$ | \( 1 - T - 93 T^{2} + 24 T^{3} + 5296 T^{4} + 24 p T^{5} - 93 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ab_adp_y_hvs |
| 61 | $D_{4}$ | \( ( 1 - 19 T + 188 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.abm_bcj_anzy_eylo |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 10 T^{2} + 528 T^{3} - 4785 T^{4} + 528 p T^{5} - 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ag_ak_ui_ahcb |
| 71 | $D_4\times C_2$ | \( 1 - 5 T - 99 T^{2} + 90 T^{3} + 8560 T^{4} + 90 p T^{5} - 99 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.af_adv_dm_mrg |
| 73 | $D_4\times C_2$ | \( 1 - 13 T + 5 T^{2} - 234 T^{3} + 9230 T^{4} - 234 p T^{5} + 5 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.an_f_aja_nra |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) | 4.79.af_db_bou_ahvw |
| 83 | $C_2^2$ | \( ( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.m_acg_qq_bgmt |
| 89 | $D_{4}$ | \( ( 1 + 17 T + 226 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bi_bcn_pvy_gtci |
| 97 | $D_{4}$ | \( ( 1 + 15 T + 226 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.be_bab_ois_glzc |
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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
−7.993501936155727929224223702210, −7.936569090081237200770361831714, −7.74610402805400982243928789770, −7.35328839442690772655642346173, −7.18779109108112101095399144720, −6.76448896791493598912108967222, −6.74653510157801272387340295005, −5.79726004704278767562358579794, −5.77817723870296404482042259802, −5.56253902827618540760329984904, −5.25459509371244060599408044133, −5.20884421801332166326545654080, −5.19512268977801431460449332679, −4.72381833816772949957870296050, −4.30918016200989616783784133325, −3.91395765428257144907967478924, −3.64899686367562349721175876404, −3.63618953659879723517234296610, −3.44063337915542185001376725258, −3.00725726889165323593852849790, −2.67009180153872785290152770198, −1.95234936830704850753151687512, −1.37764538744162485237260439653, −1.08665027297194875202601513417, −0.54341911723181272273844315387,
0.54341911723181272273844315387, 1.08665027297194875202601513417, 1.37764538744162485237260439653, 1.95234936830704850753151687512, 2.67009180153872785290152770198, 3.00725726889165323593852849790, 3.44063337915542185001376725258, 3.63618953659879723517234296610, 3.64899686367562349721175876404, 3.91395765428257144907967478924, 4.30918016200989616783784133325, 4.72381833816772949957870296050, 5.19512268977801431460449332679, 5.20884421801332166326545654080, 5.25459509371244060599408044133, 5.56253902827618540760329984904, 5.77817723870296404482042259802, 5.79726004704278767562358579794, 6.74653510157801272387340295005, 6.76448896791493598912108967222, 7.18779109108112101095399144720, 7.35328839442690772655642346173, 7.74610402805400982243928789770, 7.936569090081237200770361831714, 7.993501936155727929224223702210
