| L(s) = 1 | − 6·13-s + 6·17-s + 4·23-s + 11·25-s + 4·29-s − 30·43-s + 7·49-s − 16·53-s + 28·61-s + 28·79-s − 4·101-s + 4·103-s − 16·107-s − 40·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.66·13-s + 1.45·17-s + 0.834·23-s + 11/5·25-s + 0.742·29-s − 4.57·43-s + 49-s − 2.19·53-s + 3.58·61-s + 3.15·79-s − 0.398·101-s + 0.394·103-s − 1.54·107-s − 3.76·113-s + 8/11·121-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 + 1.38·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.655923714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.655923714\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 76 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_al_a_cy |
| 7 | $D_4\times C_2$ | \( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.7.a_ah_a_e |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ai_a_hi |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ag_cv_ali_cvk |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_cm_aie_dek |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ae_dk_aky_flm |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adk_a_fpu |
| 37 | $D_4\times C_2$ | \( 1 - 27 T^{2} + 1964 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abb_a_cxo |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.41.a_abk_a_flu |
| 43 | $D_{4}$ | \( ( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.be_th_iaw_ckgy |
| 47 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 7444 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_aep_a_lai |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.q_gq_cnw_wre |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abk_a_kug |
| 61 | $D_{4}$ | \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.abc_tk_aixo_ddmc |
| 67 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15742 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ahc_a_xhm |
| 71 | $D_4\times C_2$ | \( 1 - 207 T^{2} + 20756 T^{4} - 207 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ahz_a_besi |
| 73 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_bg_a_hzy |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abc_we_alds_enrq |
| 83 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aho_a_bipi |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ami_a_cjfi |
| 97 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 32614 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ajk_a_bwgk |
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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
−6.56663973272642162165162911589, −6.49083422531267311649780103706, −6.45864926357889635205969156102, −5.86404619159394590824807599482, −5.50855859480688106425942763243, −5.41998665353216203785081250323, −5.25823550191147760265720406524, −5.05455336271707573896929649595, −4.96762695112950495745669876255, −4.70349036536317583575687078665, −4.59153717962696794662305695725, −4.03065762355027308747164910572, −4.02777993189891491782365708955, −3.57386563942310045777182783852, −3.38690996426272490826926904327, −3.21264334872218690900545378011, −2.94131362120832157522131717645, −2.64155128804851092594943799162, −2.59554790396441503191610691221, −2.06068165756576995417378386790, −1.87049109018256597770189845082, −1.32508606747883888322352882164, −1.31201403542555050136527497524, −0.63185590049222631617486496413, −0.45496962639928120241791814651,
0.45496962639928120241791814651, 0.63185590049222631617486496413, 1.31201403542555050136527497524, 1.32508606747883888322352882164, 1.87049109018256597770189845082, 2.06068165756576995417378386790, 2.59554790396441503191610691221, 2.64155128804851092594943799162, 2.94131362120832157522131717645, 3.21264334872218690900545378011, 3.38690996426272490826926904327, 3.57386563942310045777182783852, 4.02777993189891491782365708955, 4.03065762355027308747164910572, 4.59153717962696794662305695725, 4.70349036536317583575687078665, 4.96762695112950495745669876255, 5.05455336271707573896929649595, 5.25823550191147760265720406524, 5.41998665353216203785081250323, 5.50855859480688106425942763243, 5.86404619159394590824807599482, 6.45864926357889635205969156102, 6.49083422531267311649780103706, 6.56663973272642162165162911589
