| L(s) = 1 | + 3-s + 5-s + 2·7-s − 9-s − 4·11-s + 13-s + 15-s + 11·17-s + 2·21-s − 23-s − 10·25-s + 4·27-s + 3·29-s − 6·31-s − 4·33-s + 2·35-s + 12·37-s + 39-s + 19·41-s − 5·43-s − 45-s − 17·47-s − 3·49-s + 11·51-s + 5·53-s − 4·55-s + 13·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s + 2.66·17-s + 0.436·21-s − 0.208·23-s − 2·25-s + 0.769·27-s + 0.557·29-s − 1.07·31-s − 0.696·33-s + 0.338·35-s + 1.97·37-s + 0.160·39-s + 2.96·41-s − 0.762·43-s − 0.149·45-s − 2.47·47-s − 3/7·49-s + 1.54·51-s + 0.686·53-s − 0.539·55-s + 1.69·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.475015467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.475015467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} - 7 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_c_ah |
| 5 | $S_4\times C_2$ | \( 1 - T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ab_l_ai |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_h_e |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bi_dg |
| 13 | $S_4\times C_2$ | \( 1 - T + 7 T^{2} + 50 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ab_h_by |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 75 T^{2} - 342 T^{3} + 75 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.al_cx_ane |
| 23 | $S_4\times C_2$ | \( 1 + T + 65 T^{2} + 44 T^{3} + 65 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.23.b_cn_bs |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 176 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_bv_agu |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 39 T^{2} + 156 T^{3} + 39 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.g_bn_ga |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 93 T^{2} - 596 T^{3} + 93 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.am_dp_awy |
| 41 | $S_4\times C_2$ | \( 1 - 19 T + 214 T^{2} - 39 p T^{3} + 214 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.at_ig_acjn |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 45 T^{2} + 2 p T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_bt_di |
| 47 | $S_4\times C_2$ | \( 1 + 17 T + 3 p T^{2} + 876 T^{3} + 3 p^{2} T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.r_fl_bhs |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 151 T^{2} - 486 T^{3} + 151 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.af_fv_ass |
| 59 | $S_4\times C_2$ | \( 1 - 13 T + 226 T^{2} - 1587 T^{3} + 226 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.an_is_acjb |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} + 608 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.d_df_xk |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 170 T^{2} + 1183 T^{3} + 170 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.j_go_btn |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 209 T^{2} + 422 T^{3} + 209 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.d_ib_qg |
| 73 | $S_4\times C_2$ | \( 1 + 11 T + 242 T^{2} + 1587 T^{3} + 242 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.l_ji_cjb |
| 79 | $S_4\times C_2$ | \( 1 + 19 T + 325 T^{2} + 2986 T^{3} + 325 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.t_mn_ekw |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 206 T^{2} + 1360 T^{3} + 206 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.m_hy_cai |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} - 10 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.d_df_ak |
| 97 | $S_4\times C_2$ | \( 1 + T + 274 T^{2} + 193 T^{3} + 274 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.97.b_ko_hl |
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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
−7.35716247255242385010597910709, −7.05680246194221762492855941895, −6.60734503862413339951265177390, −6.55436806973240187178886148605, −5.92501361041940232343089202547, −5.90893856873188422776156568800, −5.69288078669495446141632097644, −5.65661031676356011739636495880, −5.31608318336868818742813334225, −5.09098426285716694874617277682, −4.62821123213459018542423875210, −4.43029738611571256098060941847, −4.26084985920272053382126073611, −4.00317350399981763082594013997, −3.50968936680920032163020032786, −3.36844968224292239623155878532, −2.86188055808410258132021066153, −2.85793770347300422160374823234, −2.73368562577039639164989952449, −2.17389859095709820644644865270, −1.82240106483213728554632379380, −1.51662505052580327591877854226, −1.36970440376529965753654170298, −0.65951116488570656497468819655, −0.49136921366460120001382736328,
0.49136921366460120001382736328, 0.65951116488570656497468819655, 1.36970440376529965753654170298, 1.51662505052580327591877854226, 1.82240106483213728554632379380, 2.17389859095709820644644865270, 2.73368562577039639164989952449, 2.85793770347300422160374823234, 2.86188055808410258132021066153, 3.36844968224292239623155878532, 3.50968936680920032163020032786, 4.00317350399981763082594013997, 4.26084985920272053382126073611, 4.43029738611571256098060941847, 4.62821123213459018542423875210, 5.09098426285716694874617277682, 5.31608318336868818742813334225, 5.65661031676356011739636495880, 5.69288078669495446141632097644, 5.90893856873188422776156568800, 5.92501361041940232343089202547, 6.55436806973240187178886148605, 6.60734503862413339951265177390, 7.05680246194221762492855941895, 7.35716247255242385010597910709
