| L(s) = 1 | − 24·23-s + 2·25-s − 24·47-s − 28·49-s − 48·71-s − 52·73-s − 8·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.00·23-s + 2/5·25-s − 3.50·47-s − 4·49-s − 5.69·71-s − 6.08·73-s − 0.812·97-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ac_a_bz |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_au_a_ne |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.13.a_c_a_nb |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_ck_a_chf |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abs_a_buk |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.23.y_lw_dsy_vic |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adu_a_gbb |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ado_a_fzi |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.37.a_adq_a_hih |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_cq_a_gru |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_agi_a_pkw |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.47.y_po_ghk_bzoo |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afk_a_poo |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_afk_a_rog |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ahi_a_yjj |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_afk_a_unu |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.71.bw_bse_zjc_jwxq |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) | 4.73.ca_byg_bdwa_lwoh |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_dw_a_weg |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aem_a_zji |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_lq_a_cfeh |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.97.i_pw_dmu_dmla |
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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
−6.08890622526404127573321871251, −6.07789309538442546985673360933, −5.92002649280080960154492075158, −5.70311865303312156974552125663, −5.54729589153675826110784237795, −5.05151784135501479440344033942, −4.95878510409000601150796491451, −4.86373928646074752124097278081, −4.57963483915091943251316250825, −4.30431105312501041645212450701, −4.25840228176335216253718094460, −4.13210561602851606595226075950, −4.00619475227114442642639365592, −3.40624226006313493139765982421, −3.29348057230454880767960042360, −3.28931046303463703379564698645, −3.06197584669341828420691229018, −2.78476856436064113206345388634, −2.56144181209506440752590237898, −2.04876814646362710055057773879, −2.03386639888609336429578159708, −1.66975673065966693013489266409, −1.63525616309888232568978882135, −1.24929201275612132515957767737, −1.24403584584255264051726126025, 0, 0, 0, 0,
1.24403584584255264051726126025, 1.24929201275612132515957767737, 1.63525616309888232568978882135, 1.66975673065966693013489266409, 2.03386639888609336429578159708, 2.04876814646362710055057773879, 2.56144181209506440752590237898, 2.78476856436064113206345388634, 3.06197584669341828420691229018, 3.28931046303463703379564698645, 3.29348057230454880767960042360, 3.40624226006313493139765982421, 4.00619475227114442642639365592, 4.13210561602851606595226075950, 4.25840228176335216253718094460, 4.30431105312501041645212450701, 4.57963483915091943251316250825, 4.86373928646074752124097278081, 4.95878510409000601150796491451, 5.05151784135501479440344033942, 5.54729589153675826110784237795, 5.70311865303312156974552125663, 5.92002649280080960154492075158, 6.07789309538442546985673360933, 6.08890622526404127573321871251
