| L(s) = 1 | − 2·3-s − 4·5-s − 2·7-s + 9-s + 2·11-s + 8·15-s + 4·17-s + 6·19-s + 4·21-s + 2·23-s − 10·25-s + 2·27-s − 6·29-s − 8·31-s − 4·33-s + 8·35-s + 12·37-s + 16·41-s − 2·43-s − 4·45-s − 20·47-s + 2·49-s − 8·51-s + 12·53-s − 8·55-s − 12·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 2.06·15-s + 0.970·17-s + 1.37·19-s + 0.872·21-s + 0.417·23-s − 2·25-s + 0.384·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 1.35·35-s + 1.97·37-s + 2.49·41-s − 0.304·43-s − 0.596·45-s − 2.91·47-s + 2/7·49-s − 1.12·51-s + 1.64·53-s − 1.07·55-s − 1.58·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7147767608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7147767608\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.5.e_ba_cm_id |
| 7 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.c_c_ay_acv |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 6 T^{2} + 24 T^{3} - 65 T^{4} + 24 p T^{5} - 6 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ac_ag_y_acn |
| 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 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 24 T^{3} + 111 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_c_y_eh |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 30 T^{2} + 24 T^{3} + 535 T^{4} + 24 p T^{5} - 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ac_abe_y_up |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.g_abf_cc_dwe |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.i_bs_nw_epq |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 47 T^{2} - 276 T^{3} + 2712 T^{4} - 276 p T^{5} + 47 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.am_bv_akq_eai |
| 41 | $D_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 816 T^{3} + 5512 T^{4} - 816 p T^{5} + 3 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.aq_et_abfk_iea |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 70 T^{2} - 24 T^{3} + 3455 T^{4} - 24 p T^{5} - 70 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.c_acs_ay_fcx |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.u_ma_ens_blbq |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) | 4.53.am_kg_aczo_bhnn |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 54 T^{2} - 192 T^{3} + 475 T^{4} - 192 p T^{5} - 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.e_acc_ahk_sh |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - p T^{2} - 24 T^{3} + 8000 T^{4} - 24 p T^{5} - p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ai_acj_ay_lvs |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 26 T^{2} - 504 T^{3} + 11279 T^{4} - 504 p T^{5} + 26 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ao_ba_atk_qrv |
| 71 | $D_4\times C_2$ | \( 1 - 26 T + 378 T^{2} - 4056 T^{3} + 35767 T^{4} - 4056 p T^{5} + 378 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.aba_oo_agaa_caxr |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) | 4.73.abc_wo_alcq_ekjb |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) | 4.79.abw_btk_abbbk_lcag |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.e_ea_um_zbq |
| 89 | $D_4\times C_2$ | \( 1 - 4 T + 42 T^{2} + 816 T^{3} - 8669 T^{4} + 816 p T^{5} + 42 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ae_bq_bfk_amvl |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 94 T^{2} - 288 T^{3} + 9347 T^{4} - 288 p T^{5} - 94 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.i_adq_alc_nvn |
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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
−8.151157289488149513842841678272, −8.140783541663870027642441129047, −7.974371592106802000148146193456, −7.72525347648508254982047923444, −7.62935451291313892241780289312, −7.23508117730061127538448785988, −6.93087835010702132098129200340, −6.61746813186597095788143631244, −6.52181702791632764589740287631, −6.10071700097751240563718355436, −5.82671297125632708429094782304, −5.59533627789894675940239153603, −5.39458720731374372812390388586, −4.96097792207926246226298833518, −4.92879554675441559601702054416, −4.28141050515175846544284002399, −3.82434984842251552911617492172, −3.75890719590719082417802568656, −3.66412426494240923247692663396, −3.42531735867207207415902522980, −2.76744868603399712662378713693, −2.14884726539229334641773587923, −1.94892222844565121960075141647, −0.73283795146357341993373218473, −0.67953020918653633585030307650,
0.67953020918653633585030307650, 0.73283795146357341993373218473, 1.94892222844565121960075141647, 2.14884726539229334641773587923, 2.76744868603399712662378713693, 3.42531735867207207415902522980, 3.66412426494240923247692663396, 3.75890719590719082417802568656, 3.82434984842251552911617492172, 4.28141050515175846544284002399, 4.92879554675441559601702054416, 4.96097792207926246226298833518, 5.39458720731374372812390388586, 5.59533627789894675940239153603, 5.82671297125632708429094782304, 6.10071700097751240563718355436, 6.52181702791632764589740287631, 6.61746813186597095788143631244, 6.93087835010702132098129200340, 7.23508117730061127538448785988, 7.62935451291313892241780289312, 7.72525347648508254982047923444, 7.974371592106802000148146193456, 8.140783541663870027642441129047, 8.151157289488149513842841678272
