| L(s) = 1 | − 15·13-s + 3·19-s + 9·27-s − 39·43-s − 42·61-s + 8·64-s + 33·67-s + 51·73-s + 12·79-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 111·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4.16·13-s + 0.688·19-s + 1.73·27-s − 5.94·43-s − 5.37·61-s + 64-s + 4.03·67-s + 5.96·73-s + 1.35·79-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3806674620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3806674620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 - p^{2} T^{3} + p^{3} T^{6} \) | |
| 19 | \( ( 1 - T + p T^{2} )^{3} \) | |
| good | 2 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.2.a_a_a_a_a_ai |
| 5 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) | 6.5.a_a_a_a_a_ev |
| 7 | \( ( 1 + 17 T^{3} + p^{3} T^{6} )^{2} \) | 6.7.a_a_bi_a_a_bln |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \) | 6.11.a_bh_a_bby_a_nuj |
| 13 | \( ( 1 + 5 T + p T^{2} )^{3}( 1 - 89 T^{3} + p^{3} T^{6} ) \) | 6.13.p_ek_qk_fr_algt_acjhx |
| 17 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) | 6.17.a_a_a_a_a_hgz |
| 23 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) | 6.23.a_a_a_a_a_rzz |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.29.a_a_a_a_a_abkcb |
| 31 | \( ( 1 - 19 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \) | 6.31.a_a_a_a_a_djpt |
| 37 | \( ( 1 - 323 T^{3} + p^{3} T^{6} )( 1 + 323 T^{3} + p^{3} T^{6} ) \) | 6.37.a_a_a_a_a_aemh |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.41.a_a_a_a_a_adxyv |
| 43 | \( ( 1 + 13 T + p T^{2} )^{3}( 1 - 449 T^{3} + p^{3} T^{6} ) \) | 6.43.bn_ym_hog_ooj_amdtt_afcttp |
| 47 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) | 6.47.a_a_a_a_a_fxpf |
| 53 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.53.a_a_a_a_a_aimgb |
| 59 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.59.a_a_a_a_a_alrvf |
| 61 | \( ( 1 + 14 T + p T^{2} )^{3}( 1 - 901 T^{3} + p^{3} T^{6} ) \) | 6.61.bq_bdr_khz_npl_abeqlh_aonngk |
| 67 | \( ( 1 - 11 T + p T^{2} )^{3}( 1 - 127 T^{3} + p^{3} T^{6} ) \) | 6.67.abh_vs_aise_ckcp_amncn_cxuqv |
| 71 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.71.a_a_a_a_a_aujlv |
| 73 | \( ( 1 - 17 T + p T^{2} )^{3}( 1 + 919 T^{3} + p^{3} T^{6} ) \) | 6.73.abz_bpu_aqya_bvyn_bpiix_axdyst |
| 79 | \( ( 1 - 4 T + p T^{2} )^{3}( 1 - 503 T^{3} + p^{3} T^{6} ) \) | 6.79.am_kz_adqt_bqgd_amkwd_eifco |
| 83 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \) | 6.83.a_jp_a_cjdu_a_itswr |
| 89 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) | 6.89.a_a_a_a_a_abocwf |
| 97 | \( ( 1 + 523 T^{3} + p^{3} T^{6} )( 1 + 1853 T^{3} + p^{3} T^{6} ) \) | 6.97.a_a_dnk_a_a_gczvl |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.018146396796675506732808561040, −8.350873849236777144895679603504, −8.263006175178585232216631542131, −8.064929316675551987305439998889, −8.032380330471108314647794843900, −7.88770969467405193721250859851, −7.37409331340862537581677804791, −7.11878001837243104876508638862, −6.97914069308593375467745359082, −6.63803538154464438987919619282, −6.63495525779741110114564311192, −6.51522301149923945587297386515, −5.98448559415365132813079621886, −5.35164250359100308070372905316, −5.20493080529604940691327026659, −5.18183008782740260463291089300, −4.85676671350383420123401180070, −4.72610136053728018033364003581, −4.48521185472642810990483918005, −3.78646977919185886455903144281, −3.32614748604183387267011342222, −3.26047994195234543095404170926, −2.74236775234105993300540668983, −2.20164391801485012388063492124, −1.96527678731675760254305012060,
1.96527678731675760254305012060, 2.20164391801485012388063492124, 2.74236775234105993300540668983, 3.26047994195234543095404170926, 3.32614748604183387267011342222, 3.78646977919185886455903144281, 4.48521185472642810990483918005, 4.72610136053728018033364003581, 4.85676671350383420123401180070, 5.18183008782740260463291089300, 5.20493080529604940691327026659, 5.35164250359100308070372905316, 5.98448559415365132813079621886, 6.51522301149923945587297386515, 6.63495525779741110114564311192, 6.63803538154464438987919619282, 6.97914069308593375467745359082, 7.11878001837243104876508638862, 7.37409331340862537581677804791, 7.88770969467405193721250859851, 8.032380330471108314647794843900, 8.064929316675551987305439998889, 8.263006175178585232216631542131, 8.350873849236777144895679603504, 9.018146396796675506732808561040
Plot not available for L-functions of degree greater than 10.
