| L(s) = 1 | + 4·2-s + 10·4-s + 5·7-s + 20·8-s − 3·11-s + 8·13-s + 20·14-s + 35·16-s + 3·17-s + 5·19-s − 12·22-s − 3·23-s − 2·25-s + 32·26-s + 50·28-s + 11·31-s + 56·32-s + 12·34-s + 11·37-s + 20·38-s + 12·41-s + 11·43-s − 30·44-s − 12·46-s − 3·47-s − 8·50-s + 80·52-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.88·7-s + 7.07·8-s − 0.904·11-s + 2.21·13-s + 5.34·14-s + 35/4·16-s + 0.727·17-s + 1.14·19-s − 2.55·22-s − 0.625·23-s − 2/5·25-s + 6.27·26-s + 9.44·28-s + 1.97·31-s + 9.89·32-s + 2.05·34-s + 1.80·37-s + 3.24·38-s + 1.87·41-s + 1.67·43-s − 4.52·44-s − 1.76·46-s − 0.437·47-s − 1.13·50-s + 11.0·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 113^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 113^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(74.12691493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(74.12691493\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 3 | | \( 1 \) | |
| 113 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 6 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_c_a_ag |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 25 T^{2} - 89 T^{3} + 244 T^{4} - 89 p T^{5} + 25 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.af_z_adl_jk |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 35 T^{2} + 75 T^{3} + 516 T^{4} + 75 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.d_bj_cx_tw |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.13.ai_cy_ang_clq |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + p T^{2} - 9 T^{3} + 252 T^{4} - 9 p T^{5} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ad_r_aj_js |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 73 T^{2} - 269 T^{3} + 2044 T^{4} - 269 p T^{5} + 73 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.af_cv_akj_daq |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 59 T^{2} + 261 T^{3} + 1638 T^{4} + 261 p T^{5} + 59 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.d_ch_kb_cla |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 98 T^{2} + 4026 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_du_a_fyw |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 79 T^{2} - 431 T^{3} + 2752 T^{4} - 431 p T^{5} + 79 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.al_db_aqp_ebw |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 139 T^{2} - 827 T^{3} + 6550 T^{4} - 827 p T^{5} + 139 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.al_fj_abfv_jry |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.am_ea_abiq_lbi |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 163 T^{2} - 1139 T^{3} + 10120 T^{4} - 1139 p T^{5} + 163 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.al_gh_abrv_ozg |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 101 T^{2} + 453 T^{3} + 6126 T^{4} + 453 p T^{5} + 101 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.d_dx_rl_jbq |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 131 T^{2} + 783 T^{3} + 7296 T^{4} + 783 p T^{5} + 131 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.j_fb_bed_kuq |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 40 T^{2} + 7134 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_abo_a_kok |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 100 T^{2} + 634 T^{3} + 3046 T^{4} + 634 p T^{5} + 100 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ac_dw_yk_ene |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 259 T^{2} - 1931 T^{3} + 25528 T^{4} - 1931 p T^{5} + 259 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.al_jz_acwh_bltw |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 233 T^{2} + 1575 T^{3} + 23118 T^{4} + 1575 p T^{5} + 233 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.j_iz_cip_bife |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.au_mq_afnw_cepi |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 190 T^{2} + 916 T^{3} + 19882 T^{4} + 916 p T^{5} + 190 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.e_hi_bjg_bdks |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 281 T^{2} - 603 T^{3} + 33120 T^{4} - 603 p T^{5} + 281 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ad_kv_axf_bwzw |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 209 T^{2} + 57 T^{3} + 20772 T^{4} + 57 p T^{5} + 209 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.d_ib_cf_besy |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 235 T^{2} + 1219 T^{3} + 25204 T^{4} + 1219 p T^{5} + 235 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.97.b_jb_bux_blhk |
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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
−6.20940843924914254486597865708, −6.09606602817689181597459019105, −6.04319915496777864979978011793, −6.00720199794330744049122600967, −5.46981883388045668971632215032, −5.31612109046308194418235259977, −5.27559302522512296219215330353, −4.99133681179813676323160244290, −4.86893664650880046558379593121, −4.42683291978570194221111506368, −4.35685309479972789343824312992, −4.18552083548024316279899025230, −4.16385919263295836266635105430, −3.56698857818059641897980908498, −3.42898979909167124480725673976, −3.36743799069801559201210545885, −3.09774320722437432948880225001, −2.68352627467534091963957957734, −2.32223567722885735240426504196, −2.26510202106001385840712666299, −2.21153946189950694638245602566, −1.37256480649740890275650398827, −1.26296126061945877438260142696, −1.22166555446409957663308655470, −0.75061402119049549561768321809,
0.75061402119049549561768321809, 1.22166555446409957663308655470, 1.26296126061945877438260142696, 1.37256480649740890275650398827, 2.21153946189950694638245602566, 2.26510202106001385840712666299, 2.32223567722885735240426504196, 2.68352627467534091963957957734, 3.09774320722437432948880225001, 3.36743799069801559201210545885, 3.42898979909167124480725673976, 3.56698857818059641897980908498, 4.16385919263295836266635105430, 4.18552083548024316279899025230, 4.35685309479972789343824312992, 4.42683291978570194221111506368, 4.86893664650880046558379593121, 4.99133681179813676323160244290, 5.27559302522512296219215330353, 5.31612109046308194418235259977, 5.46981883388045668971632215032, 6.00720199794330744049122600967, 6.04319915496777864979978011793, 6.09606602817689181597459019105, 6.20940843924914254486597865708
