| L(s) = 1 | + 3·5-s − 3·7-s − 3·13-s − 6·17-s + 9·19-s + 6·23-s − 6·25-s + 3·29-s + 9·31-s − 9·35-s − 15·37-s − 6·41-s + 15·43-s + 9·47-s + 6·49-s + 6·53-s + 3·59-s − 12·61-s − 9·65-s + 12·67-s + 21·71-s + 3·73-s + 15·79-s − 6·83-s − 18·85-s + 18·89-s + 9·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 0.832·13-s − 1.45·17-s + 2.06·19-s + 1.25·23-s − 6/5·25-s + 0.557·29-s + 1.61·31-s − 1.52·35-s − 2.46·37-s − 0.937·41-s + 2.28·43-s + 1.31·47-s + 6/7·49-s + 0.824·53-s + 0.390·59-s − 1.53·61-s − 1.11·65-s + 1.46·67-s + 2.49·71-s + 0.351·73-s + 1.68·79-s − 0.658·83-s − 1.95·85-s + 1.90·89-s + 0.943·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.450377415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.450377415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 29 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_p_abd |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) | 3.11.a_y_aj |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 75 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_be_cx |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 54 T^{2} + 203 T^{3} + 54 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.g_cc_hv |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 17 p T^{3} + 63 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.aj_cl_aml |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 7 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_y_ah |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 155 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_bz_afz |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 485 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_dv_asr |
| 37 | $A_4\times C_2$ | \( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 165 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.p_gj_bsx |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 108 T^{2} + 473 T^{3} + 108 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_ee_sf |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 183 T^{2} - 1273 T^{3} + 183 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ap_hb_abwz |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 147 T^{2} - 847 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.aj_fr_abgp |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 617 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_gg_axt |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} + 135 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_bh_ff |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 192 T^{2} + 1391 T^{3} + 192 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_hk_cbn |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 1281 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_ga_abxh |
| 71 | $A_4\times C_2$ | \( 1 - 21 T + 249 T^{2} - 2115 T^{3} + 249 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.av_jp_addj |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 475 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ad_eb_ash |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 195 T^{2} - 1433 T^{3} + 195 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ap_hn_acdd |
| 83 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 1539 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.g_da_chf |
| 89 | $A_4\times C_2$ | \( 1 - 18 T + 318 T^{2} - 3241 T^{3} + 318 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.as_mg_aeur |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 57 T^{2} + 311 T^{3} + 57 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ap_cf_lz |
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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
−6.84265023592385566260119570534, −6.52420132860505270256062507958, −6.42428192542730361503269565276, −6.20589568053361955392840618346, −5.92677626236596525643964033195, −5.56467107358535976174956455477, −5.54581193989269308545396616500, −5.10808174913628658183882527390, −5.10401249935893826809905281050, −4.95735880251478022352499483327, −4.38489505943158945544014373528, −4.20041969951546270606649997050, −4.09774645952002925146413941287, −3.58656676249718544358578802659, −3.30281685730568073087897900673, −3.27068266268950213905337288966, −2.89644965119238528042385376950, −2.56820976122091824400830153550, −2.34069707002746933547932828639, −2.09257618724218220383161314607, −1.76392028998598566113426962756, −1.65389287250209917126837319312, −0.70854206264898913283623119668, −0.65413206049932382237357048379, −0.65187558950093180119691289059,
0.65187558950093180119691289059, 0.65413206049932382237357048379, 0.70854206264898913283623119668, 1.65389287250209917126837319312, 1.76392028998598566113426962756, 2.09257618724218220383161314607, 2.34069707002746933547932828639, 2.56820976122091824400830153550, 2.89644965119238528042385376950, 3.27068266268950213905337288966, 3.30281685730568073087897900673, 3.58656676249718544358578802659, 4.09774645952002925146413941287, 4.20041969951546270606649997050, 4.38489505943158945544014373528, 4.95735880251478022352499483327, 5.10401249935893826809905281050, 5.10808174913628658183882527390, 5.54581193989269308545396616500, 5.56467107358535976174956455477, 5.92677626236596525643964033195, 6.20589568053361955392840618346, 6.42428192542730361503269565276, 6.52420132860505270256062507958, 6.84265023592385566260119570534
