| L(s) = 1 | + 2·3-s − 3·5-s − 7·7-s − 9-s + 3·11-s − 6·13-s − 6·15-s − 5·17-s + 7·19-s − 14·21-s + 3·23-s + 6·25-s − 7·27-s + 29-s − 10·31-s + 6·33-s + 21·35-s − 2·37-s − 12·39-s − 10·41-s + 12·43-s + 3·45-s − 47-s + 17·49-s − 10·51-s − 10·53-s − 9·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 2.64·7-s − 1/3·9-s + 0.904·11-s − 1.66·13-s − 1.54·15-s − 1.21·17-s + 1.60·19-s − 3.05·21-s + 0.625·23-s + 6/5·25-s − 1.34·27-s + 0.185·29-s − 1.79·31-s + 1.04·33-s + 3.54·35-s − 0.328·37-s − 1.92·39-s − 1.56·41-s + 1.82·43-s + 0.447·45-s − 0.145·47-s + 17/7·49-s − 1.40·51-s − 1.37·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_f_af |
| 7 | $S_4\times C_2$ | \( 1 + p T + 32 T^{2} + 102 T^{3} + 32 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.7.h_bg_dy |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 32 T^{2} - 64 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ad_bg_acm |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 157 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bv_gb |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 20 T^{2} + 22 T^{3} + 20 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.f_u_w |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 46 T^{2} - 160 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ah_bu_age |
| 29 | $S_4\times C_2$ | \( 1 - T - 3 T^{2} + 90 T^{3} - 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ab_ad_dm |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 567 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.k_ed_vv |
| 37 | $D_{6}$ | \( 1 + 2 T - 17 T^{2} - 364 T^{3} - 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_ar_aoa |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 75 T^{2} + 339 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_cx_nb |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 776 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.am_ej_abdw |
| 47 | $S_4\times C_2$ | \( 1 + T + 135 T^{2} + 90 T^{3} + 135 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.47.b_ff_dm |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 1052 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_ed_bom |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 53 T^{2} + 4 T^{3} + 53 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ak_cb_e |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 154 T^{2} - 1372 T^{3} + 154 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.an_fy_acau |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} - 380 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_bp_aoq |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 113 T^{2} + 545 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_ej_uz |
| 73 | $S_4\times C_2$ | \( 1 - 7 T + 185 T^{2} - 786 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ah_hd_abeg |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 253 T^{2} + 1988 T^{3} + 253 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.o_jt_cym |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 309 T^{2} - 2688 T^{3} + 309 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aq_lx_adzk |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 299 T^{2} + 3304 T^{3} + 299 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_ln_exc |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 268 T^{2} - 914 T^{3} + 268 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.af_ki_abje |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.27957318184704056133901414442, −7.17736331041954542395349507355, −6.88064748126211441083719371624, −6.71067251703433362236749818732, −6.56334420229582478817375143659, −6.36948580857612865478355884929, −5.93504393991789923372366009272, −5.65087954329435709629557833349, −5.44305550655658163741622836837, −5.29405609369867652902030709290, −4.90087492717399717414271066274, −4.62762429233292709234485947794, −4.38779041060809567648868268005, −3.94670688106637820858992154241, −3.90290035523541040267592767759, −3.53360930879899913626932836946, −3.32369854638856356581758360762, −3.23147174306395210483558701884, −3.03351994863704162982093697712, −2.47575935989299656273026977826, −2.43165997956044617719264518234, −2.39649366799715679796690607357, −1.56192642692405592630060284901, −1.33311487616825555007860774848, −0.793941239238585127950754489727, 0, 0, 0,
0.793941239238585127950754489727, 1.33311487616825555007860774848, 1.56192642692405592630060284901, 2.39649366799715679796690607357, 2.43165997956044617719264518234, 2.47575935989299656273026977826, 3.03351994863704162982093697712, 3.23147174306395210483558701884, 3.32369854638856356581758360762, 3.53360930879899913626932836946, 3.90290035523541040267592767759, 3.94670688106637820858992154241, 4.38779041060809567648868268005, 4.62762429233292709234485947794, 4.90087492717399717414271066274, 5.29405609369867652902030709290, 5.44305550655658163741622836837, 5.65087954329435709629557833349, 5.93504393991789923372366009272, 6.36948580857612865478355884929, 6.56334420229582478817375143659, 6.71067251703433362236749818732, 6.88064748126211441083719371624, 7.17736331041954542395349507355, 7.27957318184704056133901414442
