| L(s) = 1 | − 3-s + 3·5-s + 7-s − 4·9-s + 11·13-s − 3·15-s − 3·17-s + 3·19-s − 21-s + 9·23-s + 6·25-s + 5·27-s + 7·29-s − 6·31-s + 3·35-s + 20·37-s − 11·39-s − 22·41-s − 10·43-s − 12·45-s − 4·49-s + 3·51-s + 7·53-s − 3·57-s + 11·59-s + 16·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s − 4/3·9-s + 3.05·13-s − 0.774·15-s − 0.727·17-s + 0.688·19-s − 0.218·21-s + 1.87·23-s + 6/5·25-s + 0.962·27-s + 1.29·29-s − 1.07·31-s + 0.507·35-s + 3.28·37-s − 1.76·39-s − 3.43·41-s − 1.52·43-s − 1.78·45-s − 4/7·49-s + 0.420·51-s + 0.961·53-s − 0.397·57-s + 1.43·59-s + 2.04·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.716589485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.716589485\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 4 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_f_e |
| 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.ab_f_abe |
| 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_q |
| 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.al_cd_ahs |
| 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.d_bv_du |
| 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.g_cz_nc |
| 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.au_jd_acoi |
| 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.w_ip_cim |
| 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.al_bl_jm |
| 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.aq_hz_acjo |
| 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.ba_rd_gvc |
| 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.o_ll_dma |
| 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.g_p_ahg |
| 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
−6.95024310924526255478959768554, −6.76133110924091097130551935684, −6.75505132387456194889344230568, −6.40370113376130157016187234820, −6.11186163757255114833204963807, −5.84877752930302281723861764207, −5.77760218404292324662834329674, −5.43483391843512889863711270592, −5.27905418965772111015201611667, −5.22754637389266793592454876208, −4.75284882876662508869329710955, −4.46763682389826912635544604016, −4.27047731019441987839968445028, −3.80916713126241833901603967369, −3.66784228316154580480811045245, −3.29367484510882281022394477319, −2.88678675160056176430591975932, −2.86451459628622973401173195942, −2.68702957417201916828339275070, −2.03435629994206944625494656176, −1.80068215009896121605613098397, −1.35480773822946357624750774536, −1.22610894924815768304031703075, −0.882967278375623988755279613910, −0.37877980661816601640281592790,
0.37877980661816601640281592790, 0.882967278375623988755279613910, 1.22610894924815768304031703075, 1.35480773822946357624750774536, 1.80068215009896121605613098397, 2.03435629994206944625494656176, 2.68702957417201916828339275070, 2.86451459628622973401173195942, 2.88678675160056176430591975932, 3.29367484510882281022394477319, 3.66784228316154580480811045245, 3.80916713126241833901603967369, 4.27047731019441987839968445028, 4.46763682389826912635544604016, 4.75284882876662508869329710955, 5.22754637389266793592454876208, 5.27905418965772111015201611667, 5.43483391843512889863711270592, 5.77760218404292324662834329674, 5.84877752930302281723861764207, 6.11186163757255114833204963807, 6.40370113376130157016187234820, 6.75505132387456194889344230568, 6.76133110924091097130551935684, 6.95024310924526255478959768554
