| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·5-s + 3·6-s − 2·7-s − 10·8-s − 6·9-s − 9·10-s + 13·11-s − 6·12-s + 6·14-s − 3·15-s + 15·16-s − 17-s + 18·18-s + 3·19-s + 18·20-s + 2·21-s − 39·22-s − 10·23-s + 10·24-s + 6·25-s + 8·27-s − 12·28-s − 8·29-s + 9·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 3·4-s + 1.34·5-s + 1.22·6-s − 0.755·7-s − 3.53·8-s − 2·9-s − 2.84·10-s + 3.91·11-s − 1.73·12-s + 1.60·14-s − 0.774·15-s + 15/4·16-s − 0.242·17-s + 4.24·18-s + 0.688·19-s + 4.02·20-s + 0.436·21-s − 8.31·22-s − 2.08·23-s + 2.04·24-s + 6/5·25-s + 1.53·27-s − 2.26·28-s − 1.48·29-s + 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296493613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296493613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 13 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_h_f |
| 7 | $A_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 20 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_n_u |
| 11 | $A_4\times C_2$ | \( 1 - 13 T + 87 T^{2} - 357 T^{3} + 87 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.an_dj_ant |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 5 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.17.b_bj_f |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 53 T^{2} - 115 T^{3} + 53 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_cb_ael |
| 23 | $A_4\times C_2$ | \( 1 + 10 T + 93 T^{2} + 468 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.k_dp_sa |
| 29 | $A_4\times C_2$ | \( 1 + 8 T + 99 T^{2} + 456 T^{3} + 99 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.i_dv_ro |
| 31 | $A_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 16 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_bh_aq |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 39 T^{2} - 228 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_bn_aiu |
| 41 | $A_4\times C_2$ | \( 1 - 19 T + 213 T^{2} - 1671 T^{3} + 213 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.at_if_acmh |
| 43 | $A_4\times C_2$ | \( 1 - 5 T - 5 T^{2} + 409 T^{3} - 5 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.af_af_pt |
| 47 | $A_4\times C_2$ | \( 1 - 8 T + 97 T^{2} - 744 T^{3} + 97 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_dt_abcq |
| 53 | $A_4\times C_2$ | \( 1 + 8 T + 59 T^{2} + 280 T^{3} + 59 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.i_ch_ku |
| 59 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} + 249 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.af_br_jp |
| 61 | $A_4\times C_2$ | \( 1 + 8 T + 83 T^{2} + 408 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_df_ps |
| 67 | $A_4\times C_2$ | \( 1 - T + 59 T^{2} - 693 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ab_ch_abar |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 177 T^{2} - 1376 T^{3} + 177 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.am_gv_acay |
| 73 | $A_4\times C_2$ | \( 1 - 5 T + 113 T^{2} - 87 T^{3} + 113 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.af_ej_adj |
| 79 | $A_4\times C_2$ | \( 1 - 2 T + 173 T^{2} - 420 T^{3} + 173 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_gr_aqe |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 151 T^{2} + 1365 T^{3} + 151 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.h_fv_can |
| 89 | $A_4\times C_2$ | \( 1 - 21 T + 365 T^{2} - 3689 T^{3} + 365 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.av_ob_aflx |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 431 T^{2} - 4361 T^{3} + 431 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.av_qp_aglt |
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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
−8.619380440912888316975207591406, −7.984506507882595600222453434762, −7.78954254065772758489002267389, −7.68131644915018153669583701310, −7.35217807637898230702457402098, −6.73085890325591600980585914310, −6.53060448619932686868603863775, −6.40518576411704686942637735988, −6.36244517937276719523944557947, −6.09214815339228081427586789108, −5.67350476282372621730028534309, −5.51633917188150603268381208881, −5.49724822207970112313795540104, −4.36508051411333728301427395763, −4.36273811174029108111465395417, −4.12160487510446817401073981008, −3.30842072805832387480638305107, −3.24764000908496639215385865803, −3.13191236416241236659605913406, −2.19263776376768628408771082719, −2.03996075651733813677770914646, −2.03832017204313545327254565069, −1.21425702555452416653422497727, −0.793174046906958392505955672407, −0.60036263492667065419793626306,
0.60036263492667065419793626306, 0.793174046906958392505955672407, 1.21425702555452416653422497727, 2.03832017204313545327254565069, 2.03996075651733813677770914646, 2.19263776376768628408771082719, 3.13191236416241236659605913406, 3.24764000908496639215385865803, 3.30842072805832387480638305107, 4.12160487510446817401073981008, 4.36273811174029108111465395417, 4.36508051411333728301427395763, 5.49724822207970112313795540104, 5.51633917188150603268381208881, 5.67350476282372621730028534309, 6.09214815339228081427586789108, 6.36244517937276719523944557947, 6.40518576411704686942637735988, 6.53060448619932686868603863775, 6.73085890325591600980585914310, 7.35217807637898230702457402098, 7.68131644915018153669583701310, 7.78954254065772758489002267389, 7.984506507882595600222453434762, 8.619380440912888316975207591406
