| L(s) = 1 | − 2·4-s − 14·11-s − 5·16-s + 16·19-s + 4·29-s − 2·31-s − 4·41-s + 28·44-s + 20·59-s + 24·61-s + 20·64-s − 4·71-s − 32·76-s + 34·79-s − 20·89-s + 36·101-s − 30·109-s − 8·116-s + 95·121-s + 4·124-s + ⋯ |
| L(s) = 1 | − 4-s − 4.22·11-s − 5/4·16-s + 3.67·19-s + 0.742·29-s − 0.359·31-s − 0.624·41-s + 4.22·44-s + 2.60·59-s + 3.07·61-s + 5/2·64-s − 0.474·71-s − 3.67·76-s + 3.82·79-s − 2.11·89-s + 3.58·101-s − 2.87·109-s − 0.742·116-s + 8.63·121-s + 0.359·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.322945399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322945399\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_c_a_j |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_abi |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.o_dx_ty_cyu |
| 13 | $D_4\times C_2$ | \( 1 + 33 T^{2} + 536 T^{4} + 33 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_bh_a_uq |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bo_a_bgo |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.19.aq_gq_absy_izm |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_cq_a_dhe |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.c_ef_go_hgm |
| 37 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 950 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_bk_a_bko |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.e_ea_oa_jec |
| 43 | $D_4\times C_2$ | \( 1 + 21 T^{2} + 1952 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_v_a_cxc |
| 47 | $D_4$ | \( 1 + 37 T^{2} + 2904 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_bl_a_ehs |
| 53 | $D_4\times C_2$ | \( 1 + 193 T^{2} + 14856 T^{4} + 193 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_hl_a_vzk |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.au_mi_afaa_btra |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.61.ay_rs_ahue_cvyc |
| 67 | $D_4\times C_2$ | \( 1 + 192 T^{2} + 17006 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_hk_a_zec |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.e_iq_bbw_bhra |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ho_a_bdzi |
| 79 | $D_{4}$ | \( ( 1 - 17 T + 222 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abi_bcf_apdq_gcxw |
| 83 | $D_4$ | \( 1 - 44 T^{2} + 5814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_abs_a_ipq |
| 89 | $D_{4}$ | \( ( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.u_qy_hrg_donm |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_lg_a_chjq |
| 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
−5.48120552080137229877725468795, −5.29325737705521474315080849898, −5.19824310732573412091500449584, −5.15999206895293339618281971852, −5.15350356954839387515232197517, −4.67494874136902450626368490623, −4.52094099054741807921655320869, −4.44084017182851352240041815703, −4.32094830441444345085986095929, −3.66983941093253307083254664001, −3.62204179276230829929505403379, −3.56069062739201508784214306386, −3.38321147951061523834896543582, −2.95817786301402532013010984991, −2.93559302054979251464106063023, −2.54684714576078252056251122093, −2.41072316083348780496110180969, −2.29757543295328391514888098923, −2.23421183956313217189807488005, −1.67473372651311702172882761951, −1.50583623712680141656592111473, −0.865209013429819303733719229161, −0.76919425771496049724928448460, −0.52340628969686724374189940084, −0.31660917264444421133906290617,
0.31660917264444421133906290617, 0.52340628969686724374189940084, 0.76919425771496049724928448460, 0.865209013429819303733719229161, 1.50583623712680141656592111473, 1.67473372651311702172882761951, 2.23421183956313217189807488005, 2.29757543295328391514888098923, 2.41072316083348780496110180969, 2.54684714576078252056251122093, 2.93559302054979251464106063023, 2.95817786301402532013010984991, 3.38321147951061523834896543582, 3.56069062739201508784214306386, 3.62204179276230829929505403379, 3.66983941093253307083254664001, 4.32094830441444345085986095929, 4.44084017182851352240041815703, 4.52094099054741807921655320869, 4.67494874136902450626368490623, 5.15350356954839387515232197517, 5.15999206895293339618281971852, 5.19824310732573412091500449584, 5.29325737705521474315080849898, 5.48120552080137229877725468795
