| L(s) = 1 | + 16·13-s + 16·25-s − 8·37-s − 2·49-s − 24·61-s − 32·73-s − 64·97-s + 16·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.43·13-s + 16/5·25-s − 1.31·37-s − 2/7·49-s − 3.07·61-s − 3.74·73-s − 6.49·97-s + 1.53·109-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 + 8.30·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.745703850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.745703850\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_aq_a_ek |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_au_a_ne |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.13.aq_fs_abhw_fow |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_acm_a_cjq |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ca_a_cbu |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bc_a_bwg |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aq_a_cpe |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.37.i_gq_bjk_ouw |
| 41 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_acm_a_gms |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_abs_a_gew |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aq_a_iko |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_au_a_klq |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.61.y_rs_hue_cvyc |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_im_a_bgve |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.73.bg_baa_nki_fghq |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.79.a_ame_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_cy_a_wnm |
| 89 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_age_a_bgxm |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) | 4.97.cm_cwa_bzum_xxgw |
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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
−5.86816055360073105199745162782, −5.78677873752171513498843042968, −5.69350634602114769634069425739, −5.54824583372706696401391592506, −5.13707244051449764264724347845, −4.93411843854818119853245464653, −4.71263140818280089422446925078, −4.68210954926185838114855372776, −4.23567130977292447729165501816, −4.09799517261358728488578740385, −3.94176934696243463116602044745, −3.88321733571147502447004877214, −3.31539933004828893363612174500, −3.27829764032969069399782727756, −3.18147701850301200358326062135, −2.92382561087011068385504902401, −2.73563106964815867096470403683, −2.53869012599120926090535140241, −1.83700226315812534593332749843, −1.72930885926976960830708287200, −1.37501909149461390881464938782, −1.37315740377924290785141313479, −1.19071698470927665857351669318, −0.71085802053081255063795298137, −0.32765289180548525008186328952,
0.32765289180548525008186328952, 0.71085802053081255063795298137, 1.19071698470927665857351669318, 1.37315740377924290785141313479, 1.37501909149461390881464938782, 1.72930885926976960830708287200, 1.83700226315812534593332749843, 2.53869012599120926090535140241, 2.73563106964815867096470403683, 2.92382561087011068385504902401, 3.18147701850301200358326062135, 3.27829764032969069399782727756, 3.31539933004828893363612174500, 3.88321733571147502447004877214, 3.94176934696243463116602044745, 4.09799517261358728488578740385, 4.23567130977292447729165501816, 4.68210954926185838114855372776, 4.71263140818280089422446925078, 4.93411843854818119853245464653, 5.13707244051449764264724347845, 5.54824583372706696401391592506, 5.69350634602114769634069425739, 5.78677873752171513498843042968, 5.86816055360073105199745162782
