| L(s) = 1 | + 3·5-s − 7-s − 11·13-s + 3·17-s + 3·19-s + 9·23-s + 6·25-s + 7·29-s + 6·31-s − 3·35-s − 20·37-s + 22·41-s − 10·43-s − 4·49-s + 7·53-s − 11·59-s − 16·61-s − 33·65-s − 67-s − 5·73-s + 26·79-s + 14·83-s + 9·85-s + 6·89-s + 11·91-s + 9·95-s + 8·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 3.05·13-s + 0.727·17-s + 0.688·19-s + 1.87·23-s + 6/5·25-s + 1.29·29-s + 1.07·31-s − 0.507·35-s − 3.28·37-s + 3.43·41-s − 1.52·43-s − 4/7·49-s + 0.961·53-s − 1.43·59-s − 2.04·61-s − 4.09·65-s − 0.122·67-s − 0.585·73-s + 2.92·79-s + 1.53·83-s + 0.976·85-s + 0.635·89-s + 1.15·91-s + 0.923·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.134772251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.134772251\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.7.b_f_be |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.11.a_f_aq |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 55 T^{2} + 200 T^{3} + 55 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.l_cd_hs |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 98 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_bv_adu |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 89 T^{2} - 422 T^{3} + 89 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.aj_dl_aqg |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 43 T^{2} - 114 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ah_br_aek |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ag_cz_anc |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 237 T^{2} + 1724 T^{3} + 237 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.u_jd_coi |
| 41 | $D_{6}$ | \( 1 - 22 T + 223 T^{2} - 1572 T^{3} + 223 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aw_ip_acim |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 97 T^{2} + 508 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.k_dt_to |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} + 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) | 3.47.a_bd_ey |
| 53 | $S_4\times C_2$ | \( 1 - 7 T - 9 T^{2} + 600 T^{3} - 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ah_aj_xc |
| 59 | $S_4\times C_2$ | \( 1 + 11 T + 37 T^{2} - 246 T^{3} + 37 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.l_bl_ajm |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.q_hz_cjo |
| 67 | $S_4\times C_2$ | \( 1 + T + 101 T^{2} + 396 T^{3} + 101 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.67.b_dx_pg |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.71.a_if_a |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 47 T^{2} - 498 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.f_bv_ate |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 445 T^{2} - 4604 T^{3} + 445 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aba_rd_agvc |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 297 T^{2} - 2340 T^{3} + 297 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ao_ll_adma |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 15 T^{2} + 188 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_p_hg |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 233 T^{2} - 1260 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ai_iz_abwm |
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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
−7.16662139490473720233209085382, −6.81012129364873215901710279243, −6.49881198640548801522228587737, −6.44232972837041646887868868398, −6.02167016088007726789004800729, −5.96182014226705262708748442569, −5.60046409209438515578872106984, −5.20853437164759178589222731769, −5.09983273368408698549267846501, −4.97310923299707202040989774707, −4.70034266554946324860583680513, −4.54037681771143201405650777398, −4.41351723477041284185654964363, −3.55669838647906392106216802157, −3.48139630236974518240537941466, −3.32523158164934512220923105859, −2.86601434914876696407080710376, −2.78793156483774718438438308349, −2.45086522879100608028219007929, −2.14858829826540359927951699530, −1.97960457667849897907951243450, −1.41819110762246588444952078727, −1.20914553033951769446046501792, −0.61063418203998621476134618717, −0.46440710630750629655657171204,
0.46440710630750629655657171204, 0.61063418203998621476134618717, 1.20914553033951769446046501792, 1.41819110762246588444952078727, 1.97960457667849897907951243450, 2.14858829826540359927951699530, 2.45086522879100608028219007929, 2.78793156483774718438438308349, 2.86601434914876696407080710376, 3.32523158164934512220923105859, 3.48139630236974518240537941466, 3.55669838647906392106216802157, 4.41351723477041284185654964363, 4.54037681771143201405650777398, 4.70034266554946324860583680513, 4.97310923299707202040989774707, 5.09983273368408698549267846501, 5.20853437164759178589222731769, 5.60046409209438515578872106984, 5.96182014226705262708748442569, 6.02167016088007726789004800729, 6.44232972837041646887868868398, 6.49881198640548801522228587737, 6.81012129364873215901710279243, 7.16662139490473720233209085382
