| L(s) = 1 | − 2·3-s − 5-s − 3·7-s − 9-s − 3·13-s + 2·15-s − 14·17-s − 19-s + 6·21-s + 23-s − 3·25-s + 2·27-s + 7·29-s + 23·31-s + 3·35-s + 6·39-s + 6·41-s − 17·43-s + 45-s + 7·47-s + 6·49-s + 28·51-s − 9·53-s + 2·57-s + 26·59-s + 6·61-s + 3·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.13·7-s − 1/3·9-s − 0.832·13-s + 0.516·15-s − 3.39·17-s − 0.229·19-s + 1.30·21-s + 0.208·23-s − 3/5·25-s + 0.384·27-s + 1.29·29-s + 4.13·31-s + 0.507·35-s + 0.960·39-s + 0.937·41-s − 2.59·43-s + 0.149·45-s + 1.02·47-s + 6/7·49-s + 3.92·51-s − 1.23·53-s + 0.264·57-s + 3.38·59-s + 0.768·61-s + 0.377·63-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
| \(L(1)\) |
\(\approx\) |
\(0.4623634792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4623634792\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_k |
| 5 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.5.b_e_b |
| 11 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 26 T^{3} + 15 p T^{4} + p^{3} T^{6} \) | 3.11.a_p_ba |
| 17 | $S_4\times C_2$ | \( 1 + 14 T + 111 T^{2} + 554 T^{3} + 111 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.o_eh_vi |
| 19 | $S_4\times C_2$ | \( 1 + T + 46 T^{2} + 29 T^{3} + 46 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.19.b_bu_bd |
| 23 | $S_4\times C_2$ | \( 1 - T + 32 T^{2} + 15 T^{3} + 32 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ab_bg_p |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 66 T^{2} - 283 T^{3} + 66 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ah_co_akx |
| 31 | $S_4\times C_2$ | \( 1 - 23 T + 264 T^{2} - 1837 T^{3} + 264 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ax_ke_acsr |
| 37 | $S_4\times C_2$ | \( 1 + 93 T^{2} - 26 T^{3} + 93 p T^{4} + p^{3} T^{6} \) | 3.37.a_dp_aba |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 39 T^{2} - 20 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_bn_au |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 188 T^{2} + 1349 T^{3} + 188 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.r_hg_bzx |
| 47 | $S_4\times C_2$ | \( 1 - 7 T + 78 T^{2} - 751 T^{3} + 78 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ah_da_abcx |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 102 T^{2} + 565 T^{3} + 102 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.j_dy_vt |
| 59 | $S_4\times C_2$ | \( 1 - 26 T + 321 T^{2} - 2744 T^{3} + 321 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.aba_mj_aebo |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 99 T^{2} - 260 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ag_dv_aka |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 221 T^{2} + 1724 T^{3} + 221 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.o_in_coi |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 243 T^{2} - 1722 T^{3} + 243 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.am_jj_acog |
| 73 | $S_4\times C_2$ | \( 1 + 11 T + 154 T^{2} + 853 T^{3} + 154 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.l_fy_bgv |
| 79 | $S_4\times C_2$ | \( 1 - 17 T + 252 T^{2} - 2137 T^{3} + 252 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ar_js_adef |
| 83 | $S_4\times C_2$ | \( 1 - 11 T + 152 T^{2} - 1965 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.al_fw_acxp |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 202 T^{2} + 1205 T^{3} + 202 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.l_hu_buj |
| 97 | $S_4\times C_2$ | \( 1 + 27 T + 498 T^{2} + 5561 T^{3} + 498 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bb_te_ifx |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.02282597303442058697173788375, −6.79960613850669101765821986906, −6.66014253195736055563352234181, −6.57108739585182993156190452342, −6.23398559675404690907921069086, −6.00996886189186937991730162408, −5.83417080896080613679331966358, −5.58761556121145181097383983961, −5.11411320672259853688039845049, −5.00250106038430367398211646976, −4.55082839014571772525968645507, −4.52645846731246417045565633293, −4.44680005201322799889114197324, −4.04654718204304856034074943699, −3.64621936798618181663071797484, −3.45396707008797809009183058784, −2.95556694078221672230354420105, −2.72942069406028757604683700220, −2.55146996467664544924431287149, −2.27535740391112718245737147499, −2.02789852727885030230457817897, −1.46011732376549009015897945586, −0.76418421707751529684571866146, −0.61707154789122146109206711340, −0.22043162522555318613856830298,
0.22043162522555318613856830298, 0.61707154789122146109206711340, 0.76418421707751529684571866146, 1.46011732376549009015897945586, 2.02789852727885030230457817897, 2.27535740391112718245737147499, 2.55146996467664544924431287149, 2.72942069406028757604683700220, 2.95556694078221672230354420105, 3.45396707008797809009183058784, 3.64621936798618181663071797484, 4.04654718204304856034074943699, 4.44680005201322799889114197324, 4.52645846731246417045565633293, 4.55082839014571772525968645507, 5.00250106038430367398211646976, 5.11411320672259853688039845049, 5.58761556121145181097383983961, 5.83417080896080613679331966358, 6.00996886189186937991730162408, 6.23398559675404690907921069086, 6.57108739585182993156190452342, 6.66014253195736055563352234181, 6.79960613850669101765821986906, 7.02282597303442058697173788375
