| L(s) = 1 | − 3·4-s − 2·5-s − 2·7-s − 8·11-s + 8·17-s + 6·20-s + 24·23-s − 15·25-s + 6·28-s + 4·35-s − 24·43-s + 24·44-s − 38·47-s − 23·49-s + 16·55-s − 20·61-s + 15·64-s − 24·68-s − 24·73-s + 16·77-s + 4·83-s − 16·85-s − 72·92-s + 45·100-s + 8·101-s − 48·115-s − 16·119-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s − 0.755·7-s − 2.41·11-s + 1.94·17-s + 1.34·20-s + 5.00·23-s − 3·25-s + 1.13·28-s + 0.676·35-s − 3.65·43-s + 3.61·44-s − 5.54·47-s − 3.28·49-s + 2.15·55-s − 2.56·61-s + 15/8·64-s − 2.91·68-s − 2.80·73-s + 1.82·77-s + 0.439·83-s − 1.73·85-s − 7.50·92-s + 9/2·100-s + 0.796·101-s − 4.47·115-s − 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T^{2} + 9 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_d_a_j |
| 5 | $D_{4}$ | \( ( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.c_t_bc_fl |
| 7 | $D_{4}$ | \( ( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.c_bb_bo_kv |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.i_cg_jw_bnv |
| 13 | $C_2^2:C_4$ | \( 1 + 27 T^{2} + 369 T^{4} + 27 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_bb_a_of |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_de_apk_dhb |
| 23 | $D_{4}$ | \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ay_lm_adoi_udn |
| 29 | $C_2^2:C_4$ | \( 1 + 16 T^{2} + 1246 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_q_a_bvy |
| 31 | $C_4\times C_2$ | \( 1 + 59 T^{2} + 2341 T^{4} + 59 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ch_a_dmb |
| 37 | $C_2^2:C_4$ | \( 1 + 48 T^{2} + 2814 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_bw_a_eeg |
| 41 | $C_2^2:C_4$ | \( 1 + 119 T^{2} + 6801 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_ep_a_kbp |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 117 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.y_oo_frs_bsbb |
| 47 | $D_{4}$ | \( ( 1 + 19 T + 183 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.bm_bbz_mye_echd |
| 53 | $C_2^2:C_4$ | \( 1 - 38 T^{2} + 5479 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abm_a_ict |
| 59 | $C_2^2:C_4$ | \( 1 - 104 T^{2} + 9646 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aea_a_oha |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.u_ou_gae_cgxa |
| 67 | $C_2^2:C_4$ | \( 1 + 83 T^{2} + 10549 T^{4} + 83 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_df_a_ppt |
| 71 | $C_4\times C_2$ | \( 1 + 199 T^{2} + 18781 T^{4} + 199 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_hr_a_bbuj |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.y_qc_hlw_cwql |
| 79 | $C_2^2:C_4$ | \( 1 + 146 T^{2} + 10591 T^{4} + 146 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_fq_a_prj |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ae_gw_abae_bgop |
| 89 | $C_2^2:C_4$ | \( 1 + 256 T^{2} + 29806 T^{4} + 256 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_jw_a_bsck |
| 97 | $C_2^2:C_4$ | \( 1 + 228 T^{2} + 26694 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_iu_a_bnms |
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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
−6.57893654462552042188781147993, −6.51726079418599734061498730987, −5.86661995163424643678092228342, −5.85046023662720079549115292373, −5.83820073507639747684458537449, −5.28428876171382602404451082025, −5.18272585040832736454743164387, −5.03344743239140268126007866552, −4.98706838589486108943793947928, −4.73522509123773040839968683909, −4.51733432225526173868594556666, −4.50455178983022977095793581063, −4.27877964022742745734876731162, −3.44619558742555555209945752592, −3.43101414048794043506986659441, −3.37917083017988404068039173719, −3.26510144247503896915642680436, −3.26221579717063236169472769161, −2.96614586012797172846758164628, −2.42983823881104555334837911186, −2.34027694510851290299675886127, −1.71455338472824820316283796280, −1.60900882976105942316156812440, −1.28392907614517624756559696529, −1.04037711747277045353466480018, 0, 0, 0, 0,
1.04037711747277045353466480018, 1.28392907614517624756559696529, 1.60900882976105942316156812440, 1.71455338472824820316283796280, 2.34027694510851290299675886127, 2.42983823881104555334837911186, 2.96614586012797172846758164628, 3.26221579717063236169472769161, 3.26510144247503896915642680436, 3.37917083017988404068039173719, 3.43101414048794043506986659441, 3.44619558742555555209945752592, 4.27877964022742745734876731162, 4.50455178983022977095793581063, 4.51733432225526173868594556666, 4.73522509123773040839968683909, 4.98706838589486108943793947928, 5.03344743239140268126007866552, 5.18272585040832736454743164387, 5.28428876171382602404451082025, 5.83820073507639747684458537449, 5.85046023662720079549115292373, 5.86661995163424643678092228342, 6.51726079418599734061498730987, 6.57893654462552042188781147993
