| L(s) = 1 | + 3·3-s + 3·5-s − 3·7-s + 6·9-s + 9·11-s + 3·13-s + 9·15-s + 12·17-s + 9·19-s − 9·21-s − 6·23-s + 10·27-s + 21·29-s + 3·31-s + 27·33-s − 9·35-s + 3·37-s + 9·39-s + 3·41-s + 6·43-s + 18·45-s − 9·47-s − 3·49-s + 36·51-s + 6·53-s + 27·55-s + 27·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 1.13·7-s + 2·9-s + 2.71·11-s + 0.832·13-s + 2.32·15-s + 2.91·17-s + 2.06·19-s − 1.96·21-s − 1.25·23-s + 1.92·27-s + 3.89·29-s + 0.538·31-s + 4.70·33-s − 1.52·35-s + 0.493·37-s + 1.44·39-s + 0.468·41-s + 0.914·43-s + 2.68·45-s − 1.31·47-s − 3/7·49-s + 5.04·51-s + 0.824·53-s + 3.64·55-s + 3.57·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(36.52915297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.52915297\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 151 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_d_ab |
| 5 | $A_4\times C_2$ | \( 1 - 3 T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_j_an |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 23 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_m_x |
| 11 | $A_4\times C_2$ | \( 1 - 9 T + 48 T^{2} - 181 T^{3} + 48 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.aj_bw_agz |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 75 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_bn_acx |
| 17 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 27 p T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.am_ds_arr |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 75 T^{2} - 351 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.aj_cx_ann |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_ci_ir |
| 29 | $A_4\times C_2$ | \( 1 - 21 T + 231 T^{2} - 1539 T^{3} + 231 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.av_ix_achf |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 57 T^{2} - 237 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_cf_ajd |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 211 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_d_id |
| 41 | $A_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 27 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ad_bq_abb |
| 43 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 465 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ag_dy_arx |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 156 T^{2} + 829 T^{3} + 156 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.j_ga_bfx |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fu_ayp |
| 59 | $A_4\times C_2$ | \( 1 - 15 T + 141 T^{2} - 907 T^{3} + 141 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ap_fl_abix |
| 61 | $A_4\times C_2$ | \( 1 + 9 T - 9 T^{2} - 451 T^{3} - 9 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.j_aj_arj |
| 67 | $A_4\times C_2$ | \( 1 + 9 T + 171 T^{2} + 1169 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.j_gp_bsz |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 258 T^{2} + 2403 T^{3} + 258 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.s_jy_dol |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 186 T^{2} - 475 T^{3} + 186 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ad_he_ash |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1171 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aj_hz_abtb |
| 83 | $A_4\times C_2$ | \( 1 - 27 T + 480 T^{2} - 5111 T^{3} + 480 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.abb_sm_ahop |
| 89 | $A_4\times C_2$ | \( 1 + 24 T + 411 T^{2} + 4464 T^{3} + 411 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.y_pv_gps |
| 97 | $A_4\times C_2$ | \( 1 - 24 T + 420 T^{2} - 4835 T^{3} + 420 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ay_qe_ahdz |
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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
−6.52997108569006286072126496293, −6.44956117895305794757194672675, −6.42807801192150774376658385980, −6.21112487193392115962591532867, −6.00372330476307000354489103564, −5.78838370519701598541030273007, −5.45045226004613686665846293543, −5.17250274817165049349970109482, −4.92935167299659031799418789185, −4.58704470528293347690378363964, −4.32103143958347197565346207192, −3.99608821570342273755223194662, −3.93249297132114441975038552633, −3.40483196717777112467880927212, −3.38802848143504893174132513882, −3.30406447556239780553034185067, −2.91894442470259242688525236337, −2.68530457225517627847546645595, −2.51193949312414165788645179370, −1.87274386953852713676560068173, −1.80215325856209006878758959997, −1.27674692618299029993014873817, −1.18451481420784231310551572684, −0.955966907423714100775916369190, −0.73373470214767845996015414667,
0.73373470214767845996015414667, 0.955966907423714100775916369190, 1.18451481420784231310551572684, 1.27674692618299029993014873817, 1.80215325856209006878758959997, 1.87274386953852713676560068173, 2.51193949312414165788645179370, 2.68530457225517627847546645595, 2.91894442470259242688525236337, 3.30406447556239780553034185067, 3.38802848143504893174132513882, 3.40483196717777112467880927212, 3.93249297132114441975038552633, 3.99608821570342273755223194662, 4.32103143958347197565346207192, 4.58704470528293347690378363964, 4.92935167299659031799418789185, 5.17250274817165049349970109482, 5.45045226004613686665846293543, 5.78838370519701598541030273007, 6.00372330476307000354489103564, 6.21112487193392115962591532867, 6.42807801192150774376658385980, 6.44956117895305794757194672675, 6.52997108569006286072126496293
