| L(s) = 1 | + 2·2-s + 6·3-s + 2·4-s − 8·5-s + 12·6-s − 4·7-s + 2·8-s + 21·9-s − 16·10-s − 2·11-s + 12·12-s − 8·14-s − 48·15-s + 3·16-s − 4·17-s + 42·18-s − 16·20-s − 24·21-s − 4·22-s + 12·24-s + 26·25-s + 54·27-s − 8·28-s − 8·29-s − 96·30-s + 6·31-s + 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3.46·3-s + 4-s − 3.57·5-s + 4.89·6-s − 1.51·7-s + 0.707·8-s + 7·9-s − 5.05·10-s − 0.603·11-s + 3.46·12-s − 2.13·14-s − 12.3·15-s + 3/4·16-s − 0.970·17-s + 9.89·18-s − 3.57·20-s − 5.23·21-s − 0.852·22-s + 2.44·24-s + 26/5·25-s + 10.3·27-s − 1.51·28-s − 1.48·29-s − 17.5·30-s + 1.07·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.468663317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.468663317\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) | |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_c_ac_b |
| 5 | $C_2^2$ | \( ( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.i_bm_ey_mt |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.e_bc_cy_lm |
| 11 | $D_4\times C_2$ | \( 1 + 2 T + 2 T^{2} + 20 T^{3} + 199 T^{4} + 20 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_c_u_hr |
| 13 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 15 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_ao_a_hn |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 52 T^{3} + 322 T^{4} + 52 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_i_ca_mk |
| 19 | $C_2^3$ | \( 1 + 302 T^{4} + p^{4} T^{8} \) | 4.19.a_a_a_lq |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_aci_a_cpy |
| 31 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 204 T^{3} + 2303 T^{4} - 204 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ag_s_ahw_dkp |
| 37 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) | 4.37.a_a_a_aje |
| 41 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) | 4.41.a_a_a_adve |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 - 61 T^{2} + p^{2} T^{4} ) \) | 4.43.ak_by_gy_aeof |
| 47 | $D_4\times C_2$ | \( 1 - 14 T + 98 T^{2} - 476 T^{3} + 2143 T^{4} - 476 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ao_du_asi_del |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 6291 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_adu_a_jhz |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_agy_a_vdu |
| 61 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.i_bg_vg_nzu |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 14934 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_agi_a_wck |
| 71 | $D_{4}$ | \( ( 1 - 30 T + 364 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.aci_ckq_abmpw_pjze |
| 73 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.q_ey_cmq_bfcs |
| 79 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 540 T^{3} + 5207 T^{4} + 540 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.k_by_uu_hsh |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ca_a_vjy |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.u_hs_ecy_cbmc |
| 97 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3912 T^{3} + 48782 T^{4} + 3912 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.y_lc_fum_cueg |
| 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
−10.73126848803377284096577615456, −10.05019879989224493084680105528, −9.669609052780942083843827391482, −9.609116065850912031873943077382, −9.492457661852572178851926293300, −8.851157268663388350528539366246, −8.431649014375362330618434841101, −8.372966979678730585969204056346, −8.265616017608877052600928627605, −7.71261809984032131594904448391, −7.54754222236557277988611637100, −7.45697501640839057994477981531, −7.18644921415016326001496936084, −6.54706368023376339910757860235, −6.51465495053721839068258632211, −5.72610350175368367215024739514, −5.07798814371478233673322338487, −4.36455706855295643630641564492, −4.23761528155496304471603523463, −4.20940093610889004302015077257, −3.56578793271638368759191144750, −3.47282394528377000084870382426, −3.24335062228152316731830523739, −2.62553781196772683092613592498, −2.25748004816738255987187310989,
2.25748004816738255987187310989, 2.62553781196772683092613592498, 3.24335062228152316731830523739, 3.47282394528377000084870382426, 3.56578793271638368759191144750, 4.20940093610889004302015077257, 4.23761528155496304471603523463, 4.36455706855295643630641564492, 5.07798814371478233673322338487, 5.72610350175368367215024739514, 6.51465495053721839068258632211, 6.54706368023376339910757860235, 7.18644921415016326001496936084, 7.45697501640839057994477981531, 7.54754222236557277988611637100, 7.71261809984032131594904448391, 8.265616017608877052600928627605, 8.372966979678730585969204056346, 8.431649014375362330618434841101, 8.851157268663388350528539366246, 9.492457661852572178851926293300, 9.609116065850912031873943077382, 9.669609052780942083843827391482, 10.05019879989224493084680105528, 10.73126848803377284096577615456
