| L(s) = 1 | + 2·3-s + 4·5-s + 10·11-s − 4·13-s + 8·15-s + 6·19-s − 2·23-s + 10·25-s − 2·27-s + 4·29-s − 6·31-s + 20·33-s + 12·37-s − 8·39-s + 16·41-s − 6·43-s − 8·47-s − 8·49-s + 40·55-s + 12·57-s + 26·59-s + 4·61-s − 16·65-s − 4·67-s − 4·69-s − 34·71-s + 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 3.01·11-s − 1.10·13-s + 2.06·15-s + 1.37·19-s − 0.417·23-s + 2·25-s − 0.384·27-s + 0.742·29-s − 1.07·31-s + 3.48·33-s + 1.97·37-s − 1.28·39-s + 2.49·41-s − 0.914·43-s − 1.16·47-s − 8/7·49-s + 5.39·55-s + 1.58·57-s + 3.38·59-s + 0.512·61-s − 1.98·65-s − 0.488·67-s − 0.481·69-s − 4.03·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.20060553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.20060553\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + 10 T^{4} - 2 p^{2} T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_e_ag_k |
| 7 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} - 16 T^{3} + 30 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_i_aq_be |
| 11 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 278 T^{3} + 1018 T^{4} - 278 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ak_cm_aks_bne |
| 17 | $C_2 \wr S_4$ | \( 1 + 20 T^{2} - 64 T^{3} + 86 T^{4} - 64 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_u_acm_di |
| 19 | $C_2 \wr S_4$ | \( 1 - 6 T + 72 T^{2} - 330 T^{3} + 2010 T^{4} - 330 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_cu_ams_czi |
| 23 | $C_2 \wr S_4$ | \( 1 + 2 T + 44 T^{2} - 18 T^{3} + 842 T^{4} - 18 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_bs_as_bgk |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 92 T^{2} - 284 T^{3} + 3718 T^{4} - 284 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ae_do_aky_fna |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 64 T^{2} + 418 T^{3} + 2714 T^{4} + 418 p T^{5} + 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.g_cm_qc_eak |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 164 T^{2} - 1236 T^{3} + 9334 T^{4} - 1236 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.am_gi_abvo_nva |
| 41 | $C_2 \wr S_4$ | \( 1 - 16 T + 212 T^{2} - 1712 T^{3} + 12982 T^{4} - 1712 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.aq_ie_acnw_tfi |
| 43 | $C_2 \wr S_4$ | \( 1 + 6 T + 76 T^{2} + 418 T^{3} + 5162 T^{4} + 418 p T^{5} + 76 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.g_cy_qc_hqo |
| 47 | $C_2 \wr S_4$ | \( 1 + 8 T + 56 T^{2} + 280 T^{3} + 3550 T^{4} + 280 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.i_ce_ku_fgo |
| 53 | $C_2 \wr S_4$ | \( 1 + 164 T^{2} + 64 T^{3} + 11750 T^{4} + 64 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_gi_cm_rjy |
| 59 | $C_2 \wr S_4$ | \( 1 - 26 T + 8 p T^{2} - 5446 T^{3} + 49594 T^{4} - 5446 p T^{5} + 8 p^{3} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.aba_se_aibm_cvjm |
| 61 | $C_2 \wr S_4$ | \( 1 - 4 T + 188 T^{2} - 604 T^{3} + 16326 T^{4} - 604 p T^{5} + 188 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ae_hg_axg_ydy |
| 67 | $C_2 \wr S_4$ | \( 1 + 4 T + 200 T^{2} + 596 T^{3} + 18670 T^{4} + 596 p T^{5} + 200 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.e_hs_wy_bbqc |
| 71 | $C_2 \wr S_4$ | \( 1 + 34 T + 680 T^{2} + 8982 T^{3} + 88346 T^{4} + 8982 p T^{5} + 680 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.bi_bae_nhm_fary |
| 73 | $C_2 \wr S_4$ | \( 1 - 12 T + 228 T^{2} - 1956 T^{3} + 24230 T^{4} - 1956 p T^{5} + 228 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.am_iu_acxg_bjvy |
| 79 | $C_2 \wr S_4$ | \( 1 + 28 T + 572 T^{2} + 7500 T^{3} + 78726 T^{4} + 7500 p T^{5} + 572 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.bc_wa_lcm_emly |
| 83 | $C_2 \wr S_4$ | \( 1 - 24 T + 312 T^{2} - 2760 T^{3} + 24654 T^{4} - 2760 p T^{5} + 312 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ay_ma_aece_bkmg |
| 89 | $C_2 \wr S_4$ | \( 1 - 16 T + 300 T^{2} - 3504 T^{3} + 38150 T^{4} - 3504 p T^{5} + 300 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aq_lo_afeu_celi |
| 97 | $C_2 \wr S_4$ | \( 1 + 8 T + 300 T^{2} + 2296 T^{3} + 39462 T^{4} + 2296 p T^{5} + 300 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.i_lo_dki_cgju |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.07865527204864696076779374693, −5.65783295341163611667967024466, −5.52877984366256523710860035640, −5.52272811328431097579227147812, −5.33354407356882375033040192660, −4.82085519294587509522941924369, −4.69892188638662372071639533905, −4.52153763609525847729932423260, −4.32437221893929235362436684454, −4.20033730679226368801887504160, −3.92384614108761952132702879708, −3.71883350442086805113680327566, −3.33092011959782871033042301155, −3.12473393146542018062230401860, −3.04557559861519686607013602181, −2.95636022780165359768428033151, −2.64078989162185208620397126224, −2.19050883773324755914439746681, −2.01882991743051309190887595286, −1.97101482333108040251230507456, −1.71361476102604292013652669723, −1.36672798335419147020413509026, −0.911140769893707997974003675127, −0.911014609678005034963235882760, −0.52102809645284929837018620417,
0.52102809645284929837018620417, 0.911014609678005034963235882760, 0.911140769893707997974003675127, 1.36672798335419147020413509026, 1.71361476102604292013652669723, 1.97101482333108040251230507456, 2.01882991743051309190887595286, 2.19050883773324755914439746681, 2.64078989162185208620397126224, 2.95636022780165359768428033151, 3.04557559861519686607013602181, 3.12473393146542018062230401860, 3.33092011959782871033042301155, 3.71883350442086805113680327566, 3.92384614108761952132702879708, 4.20033730679226368801887504160, 4.32437221893929235362436684454, 4.52153763609525847729932423260, 4.69892188638662372071639533905, 4.82085519294587509522941924369, 5.33354407356882375033040192660, 5.52272811328431097579227147812, 5.52877984366256523710860035640, 5.65783295341163611667967024466, 6.07865527204864696076779374693
