| L(s) = 1 | − 5-s + 3·7-s − 5·9-s − 2·11-s + 3·13-s − 8·17-s + 7·19-s − 7·23-s − 11·25-s + 2·27-s + 7·29-s − 5·31-s − 3·35-s + 6·37-s − 2·41-s − 9·43-s + 5·45-s − 5·47-s + 6·49-s + 3·53-s + 2·55-s − 18·59-s + 14·61-s − 15·63-s − 3·65-s − 10·67-s − 2·71-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 5/3·9-s − 0.603·11-s + 0.832·13-s − 1.94·17-s + 1.60·19-s − 1.45·23-s − 2.19·25-s + 0.384·27-s + 1.29·29-s − 0.898·31-s − 0.507·35-s + 0.986·37-s − 0.312·41-s − 1.37·43-s + 0.745·45-s − 0.729·47-s + 6/7·49-s + 0.412·53-s + 0.269·55-s − 2.34·59-s + 1.79·61-s − 1.88·63-s − 0.372·65-s − 1.22·67-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.3.a_f_ac |
| 5 | $S_4\times C_2$ | \( 1 + T + 12 T^{2} + 9 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.5.b_m_j |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 54 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_t_cc |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 198 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.i_bz_hq |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 70 T^{2} - 271 T^{3} + 70 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ah_cs_akl |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 72 T^{2} + 303 T^{3} + 72 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.h_cu_lr |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 82 T^{2} - 363 T^{3} + 82 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ah_de_anz |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 36 T^{2} + 335 T^{3} + 36 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_bk_mx |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 446 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dl_are |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 103 T^{2} + 156 T^{3} + 103 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_dz_ga |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 128 T^{2} + 665 T^{3} + 128 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.j_ey_zp |
| 47 | $S_4\times C_2$ | \( 1 + 5 T + 70 T^{2} + 609 T^{3} + 70 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.f_cs_xl |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 62 T^{2} + 49 T^{3} + 62 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ad_ck_bx |
| 59 | $S_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2224 T^{3} + 257 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.s_jx_dho |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 1556 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_id_achw |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 173 T^{2} + 1332 T^{3} + 173 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.k_gr_bzg |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 155 T^{2} + 418 T^{3} + 155 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.c_fz_qc |
| 73 | $S_4\times C_2$ | \( 1 + 23 T + 346 T^{2} + 3329 T^{3} + 346 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.x_ni_eyb |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 104 T^{2} + 1467 T^{3} + 104 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.h_ea_cel |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 84 T^{2} - 1689 T^{3} + 84 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ah_dg_acmz |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 170 T^{2} + 685 T^{3} + 170 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.d_go_baj |
| 97 | $S_4\times C_2$ | \( 1 + 7 T + 2 p T^{2} + 645 T^{3} + 2 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.h_hm_yv |
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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
−7.67752530294428092518792941171, −7.38332648603975004092718120827, −6.96509743628995651732958537088, −6.94130265566182249932008121527, −6.41911416525764302970894383537, −6.39532282242798756798583795742, −6.00776417275626145589957861311, −5.77835303340414618799089715985, −5.58728296953493762022812535523, −5.53374207880008793099703776401, −5.07557411210446648765834990907, −4.77519119717132623909158186544, −4.71367347767379768943198308597, −4.17821828643055680137381164680, −4.11557853823958635811256600176, −3.99121101757346682827885309251, −3.32668875344304186358097649656, −3.29575855421198059421752846887, −3.02648995844644662359426269437, −2.50625961428216290983949957567, −2.41379908867590774887455789101, −2.13138495431091829714569841628, −1.64839597167186346075420013103, −1.30854576901265588138870827195, −1.14584537109285718151662143573, 0, 0, 0,
1.14584537109285718151662143573, 1.30854576901265588138870827195, 1.64839597167186346075420013103, 2.13138495431091829714569841628, 2.41379908867590774887455789101, 2.50625961428216290983949957567, 3.02648995844644662359426269437, 3.29575855421198059421752846887, 3.32668875344304186358097649656, 3.99121101757346682827885309251, 4.11557853823958635811256600176, 4.17821828643055680137381164680, 4.71367347767379768943198308597, 4.77519119717132623909158186544, 5.07557411210446648765834990907, 5.53374207880008793099703776401, 5.58728296953493762022812535523, 5.77835303340414618799089715985, 6.00776417275626145589957861311, 6.39532282242798756798583795742, 6.41911416525764302970894383537, 6.94130265566182249932008121527, 6.96509743628995651732958537088, 7.38332648603975004092718120827, 7.67752530294428092518792941171
