| L(s) = 1 | + 2·3-s + 4-s + 4·9-s + 2·12-s + 14·13-s − 6·17-s − 12·19-s − 12·23-s − 2·25-s + 4·27-s + 12·29-s + 4·36-s + 18·37-s + 28·39-s − 36·41-s − 4·43-s − 5·49-s − 12·51-s + 14·52-s − 12·53-s − 24·57-s + 36·59-s − 2·61-s − 64-s − 6·68-s − 24·69-s − 4·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 4/3·9-s + 0.577·12-s + 3.88·13-s − 1.45·17-s − 2.75·19-s − 2.50·23-s − 2/5·25-s + 0.769·27-s + 2.22·29-s + 2/3·36-s + 2.95·37-s + 4.48·39-s − 5.62·41-s − 0.609·43-s − 5/7·49-s − 1.68·51-s + 1.94·52-s − 1.64·53-s − 3.17·57-s + 4.68·59-s − 0.256·61-s − 1/8·64-s − 0.727·68-s − 2.88·69-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.071068447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.071068447\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_a_e_af |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_f_a_ay |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_n_a_bw |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 20 T^{2} - 108 T^{3} - 645 T^{4} - 108 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.g_u_aee_ayv |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 89 T^{2} + 492 T^{3} + 2232 T^{4} + 492 p T^{5} + 89 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.m_dl_sy_dhw |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} + 288 T^{3} + 1059 T^{4} + 288 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.m_cw_lc_bot |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} - 288 T^{3} + 1707 T^{4} - 288 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.am_ck_alc_cnr |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adw_a_gjy |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 173 T^{2} - 1170 T^{3} + 6852 T^{4} - 1170 p T^{5} + 173 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.as_gr_abta_kdo |
| 41 | $C_2^2$ | \( ( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.bk_xy_kdc_czdb |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.e_acw_q_ifz |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_ago_a_rfv |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_ji_cua_bdqx |
| 59 | $C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.abk_zi_amay_eecx |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 92 T^{2} - 52 T^{3} + 5251 T^{4} - 52 p T^{5} - 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.c_ado_aca_htz |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_fe_a_txz |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ec_a_jeh |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11802 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_aeu_a_rly |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ae_ki_abgm_bteo |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aka_a_bryo |
| 89 | $D_4\times C_2$ | \( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 5412 T^{4} + 1098 p T^{5} + 169 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.s_gn_bqg_iae |
| 97 | $D_4\times C_2$ | \( 1 - 42 T + 920 T^{2} - 13944 T^{3} + 157851 T^{4} - 13944 p T^{5} + 920 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abq_bjk_auqi_iznf |
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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
−10.15754233316880493629831271641, −9.413540063681346633807313250126, −9.177437026845674059168669774180, −8.811301885522161806246933870043, −8.494171274888494869114445454004, −8.378209864678558874944246394307, −8.255465819569279081369416427980, −8.052891047627317324138043260733, −7.921876494092038646709983613602, −7.02761420502294060023201411231, −6.61392585311012895962959764281, −6.60775075236474066458397663898, −6.50324869531993387509419249081, −6.10361021065949761062404583891, −5.99333068932491202487180354626, −5.13314679161140245856225262785, −4.95107123783892681423417044235, −4.11243522165841246154540440590, −4.02166217655859565416862422534, −4.01592167938776934834061317335, −3.53290408299305136501776025453, −2.87863066665948351263053362496, −2.39557014566654468148346202004, −1.73049891669617410671528272357, −1.64170616099128021979535656547,
1.64170616099128021979535656547, 1.73049891669617410671528272357, 2.39557014566654468148346202004, 2.87863066665948351263053362496, 3.53290408299305136501776025453, 4.01592167938776934834061317335, 4.02166217655859565416862422534, 4.11243522165841246154540440590, 4.95107123783892681423417044235, 5.13314679161140245856225262785, 5.99333068932491202487180354626, 6.10361021065949761062404583891, 6.50324869531993387509419249081, 6.60775075236474066458397663898, 6.61392585311012895962959764281, 7.02761420502294060023201411231, 7.921876494092038646709983613602, 8.052891047627317324138043260733, 8.255465819569279081369416427980, 8.378209864678558874944246394307, 8.494171274888494869114445454004, 8.811301885522161806246933870043, 9.177437026845674059168669774180, 9.413540063681346633807313250126, 10.15754233316880493629831271641
