| L(s) = 1 | + 2-s + 5·3-s + 5·5-s + 5·6-s + 13·9-s + 5·10-s + 13-s + 25·15-s − 3·17-s + 13·18-s + 8·19-s − 9·23-s + 10·25-s + 26-s + 30·27-s − 3·29-s + 25·30-s + 9·31-s − 32-s − 3·34-s + 16·37-s + 8·38-s + 5·39-s − 5·41-s + 20·43-s + 65·45-s − 9·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.88·3-s + 2.23·5-s + 2.04·6-s + 13/3·9-s + 1.58·10-s + 0.277·13-s + 6.45·15-s − 0.727·17-s + 3.06·18-s + 1.83·19-s − 1.87·23-s + 2·25-s + 0.196·26-s + 5.77·27-s − 0.557·29-s + 4.56·30-s + 1.61·31-s − 0.176·32-s − 0.514·34-s + 2.63·37-s + 1.29·38-s + 0.800·39-s − 0.780·41-s + 3.04·43-s + 9.68·45-s − 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.59544563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.59544563\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) | |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) | |
| good | 3 | $C_4\times C_2$ | \( 1 - 5 T + 4 p T^{2} - 25 T^{3} + 49 T^{4} - 25 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.af_m_az_bx |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^2:C_4$ | \( 1 - T^{2} + 30 T^{3} + 91 T^{4} + 30 p T^{5} - p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ab_be_dn |
| 17 | $C_4\times C_2$ | \( 1 + 3 T - 8 T^{2} - 75 T^{3} - 89 T^{4} - 75 p T^{5} - 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.d_ai_acx_adl |
| 19 | $C_2^2:C_4$ | \( 1 - 8 T + 15 T^{2} - 58 T^{3} + 539 T^{4} - 58 p T^{5} + 15 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_p_acg_ut |
| 23 | $C_2^2:C_4$ | \( 1 + 9 T + 38 T^{2} + 255 T^{3} + 1741 T^{4} + 255 p T^{5} + 38 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.j_bm_jv_coz |
| 29 | $C_2^2:C_4$ | \( 1 + 3 T - 10 T^{2} + 123 T^{3} + 1219 T^{4} + 123 p T^{5} - 10 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.d_ak_et_bux |
| 31 | $C_2^2:C_4$ | \( 1 - 9 T + 259 T^{3} - 1491 T^{4} + 259 p T^{5} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.aj_a_jz_acfj |
| 37 | $C_2^2:C_4$ | \( 1 - 16 T + 69 T^{2} + 88 T^{3} - 1711 T^{4} + 88 p T^{5} + 69 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.aq_cr_dk_acnv |
| 41 | $C_4\times C_2$ | \( 1 + 5 T - 26 T^{2} + 25 T^{3} + 1911 T^{4} + 25 p T^{5} - 26 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.f_aba_z_cvn |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.au_kw_adzc_beln |
| 47 | $C_2^2:C_4$ | \( 1 + 22 T + 157 T^{2} + 110 T^{3} - 3489 T^{4} + 110 p T^{5} + 157 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.w_gb_eg_afef |
| 53 | $C_2^2:C_4$ | \( 1 - 11 T - 2 T^{2} + 605 T^{3} - 4869 T^{4} + 605 p T^{5} - 2 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.al_ac_xh_ahfh |
| 59 | $C_2^2:C_4$ | \( 1 + 13 T + 10 T^{2} - 277 T^{3} - 351 T^{4} - 277 p T^{5} + 10 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.n_k_akr_ann |
| 61 | $C_2^2:C_4$ | \( 1 - 22 T + 243 T^{2} - 2384 T^{3} + 21425 T^{4} - 2384 p T^{5} + 243 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aw_jj_adns_bfsb |
| 67 | $C_2^2:C_4$ | \( 1 - 19 T + 174 T^{2} - 1793 T^{3} + 18569 T^{4} - 1793 p T^{5} + 174 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.at_gs_acqz_bbmf |
| 71 | $C_2^2:C_4$ | \( 1 + 18 T + 73 T^{2} + 6 T^{3} + 1225 T^{4} + 6 p T^{5} + 73 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.s_cv_g_bvd |
| 73 | $C_4\times C_2$ | \( 1 - 5 T - 48 T^{2} + 605 T^{3} + 479 T^{4} + 605 p T^{5} - 48 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.af_abw_xh_sl |
| 79 | $C_2^2:C_4$ | \( 1 - 13 T + 210 T^{2} - 1943 T^{3} + 24029 T^{4} - 1943 p T^{5} + 210 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.an_ic_acwt_bjof |
| 83 | $C_2^2:C_4$ | \( 1 + 13 T + 121 T^{2} + 1649 T^{3} + 21684 T^{4} + 1649 p T^{5} + 121 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.n_er_cll_bgca |
| 89 | $C_4\times C_2$ | \( 1 - 10 T - 29 T^{2} - 200 T^{3} + 10101 T^{4} - 200 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ak_abd_ahs_oyn |
| 97 | $C_2^2:C_4$ | \( 1 + 34 T + 459 T^{2} + 3488 T^{3} + 25829 T^{4} + 3488 p T^{5} + 459 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bi_rr_fee_bmfl |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68929058558953482238587386087, −7.42874722098541670384361652946, −7.21855579751183878581917958054, −6.84137324749553352648020764895, −6.50144261844547705877934814027, −6.45395925331687393973981132125, −6.35671842903869026034343622775, −5.80105128082022995396181978425, −5.71595431115394227897404078616, −5.40645522071488845239767966929, −5.31562132144636023826755873222, −4.60474953661833674303102671927, −4.52975570580274331026045641219, −4.39127543680652871585733947927, −4.30561290157365547406897610852, −3.46415552571956005993671649925, −3.43742349408634372895436869217, −3.19662484935690786532661670498, −3.07491730848479484471852305222, −2.49822978210356214366544871427, −2.29878124769409731136664273238, −2.07385980555941144911290981777, −1.98335578979385016456952805092, −1.20347569177384799162042757667, −1.09988246975896475308421806703,
1.09988246975896475308421806703, 1.20347569177384799162042757667, 1.98335578979385016456952805092, 2.07385980555941144911290981777, 2.29878124769409731136664273238, 2.49822978210356214366544871427, 3.07491730848479484471852305222, 3.19662484935690786532661670498, 3.43742349408634372895436869217, 3.46415552571956005993671649925, 4.30561290157365547406897610852, 4.39127543680652871585733947927, 4.52975570580274331026045641219, 4.60474953661833674303102671927, 5.31562132144636023826755873222, 5.40645522071488845239767966929, 5.71595431115394227897404078616, 5.80105128082022995396181978425, 6.35671842903869026034343622775, 6.45395925331687393973981132125, 6.50144261844547705877934814027, 6.84137324749553352648020764895, 7.21855579751183878581917958054, 7.42874722098541670384361652946, 7.68929058558953482238587386087
