| L(s) = 1 | − 4·2-s + 8·4-s − 4·5-s − 12·8-s + 16·10-s − 8·11-s + 15·16-s + 8·17-s + 8·19-s − 32·20-s + 32·22-s + 10·25-s − 8·29-s + 8·31-s − 16·32-s − 32·34-s + 8·37-s − 32·38-s + 48·40-s − 8·43-s − 64·44-s + 8·47-s − 40·50-s − 8·53-s + 32·55-s + 32·58-s + 16·59-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 1.78·5-s − 4.24·8-s + 5.05·10-s − 2.41·11-s + 15/4·16-s + 1.94·17-s + 1.83·19-s − 7.15·20-s + 6.82·22-s + 2·25-s − 1.48·29-s + 1.43·31-s − 2.82·32-s − 5.48·34-s + 1.31·37-s − 5.19·38-s + 7.58·40-s − 1.21·43-s − 9.64·44-s + 1.16·47-s − 5.65·50-s − 1.09·53-s + 4.31·55-s + 4.20·58-s + 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4352090471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4352090471\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 7 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) | 4.2.e_i_m_r |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 36 T^{2} + 104 T^{3} + 326 T^{4} + 104 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.i_bk_ea_mo |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T^{2} + 64 T^{3} + 198 T^{4} + 64 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_u_cm_hq |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 52 T^{2} - 280 T^{3} + 1318 T^{4} - 280 p T^{5} + 52 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ai_ca_aku_bys |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 64 T^{2} - 280 T^{3} + 1426 T^{4} - 280 p T^{5} + 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_cm_aku_ccw |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 48 T^{2} + 64 T^{3} + 1346 T^{4} + 64 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bw_cm_bzu |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 44 T^{2} + 88 T^{3} + 502 T^{4} + 88 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.i_bs_dk_ti |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 112 T^{2} - 696 T^{3} + 4994 T^{4} - 696 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ai_ei_abau_hkc |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 92 T^{2} - 600 T^{3} + 4854 T^{4} - 600 p T^{5} + 92 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_do_axc_hes |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 108 T^{2} + 128 T^{3} + 5510 T^{4} + 128 p T^{5} + 108 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_ee_ey_idy |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 140 T^{2} + 840 T^{3} + 8406 T^{4} + 840 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.i_fk_bgi_mli |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ai_cq_awm_jtu |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 208 T^{2} + 1192 T^{3} + 16402 T^{4} + 1192 p T^{5} + 208 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.i_ia_btw_ygw |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 188 T^{2} - 1424 T^{3} + 10678 T^{4} - 1424 p T^{5} + 188 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.aq_hg_accu_pus |
| 61 | $D_{4}$ | \( ( 1 - 16 T + 184 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.abg_ya_alpo_edhq |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 92 T^{2} - 512 T^{3} + 6486 T^{4} - 512 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_do_ats_jpm |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 212 T^{2} - 1608 T^{3} + 20054 T^{4} - 1608 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ai_ie_acjw_bdri |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 164 T^{2} + 128 T^{3} + 16374 T^{4} + 128 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_gi_ey_yfu |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} - 384 T^{3} + 10950 T^{4} - 384 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_fc_aou_qfe |
| 83 | $D_{4}$ | \( ( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.abo_bjg_aueq_ijby |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.i_oa_dcy_culi |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 196 T^{2} + 1280 T^{3} + 17174 T^{4} + 1280 p T^{5} + 196 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ho_bxg_zko |
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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
−6.46534334968403968896644982098, −6.40768579196158970756535205270, −6.35450128789117556701624034911, −5.59042594194994772843835997551, −5.53878511326230623713156313364, −5.41725201256928369940354094521, −5.19004240919297740875374580913, −5.15517217941081665173772338481, −5.07141341490606087655331135246, −4.30398010880638423186059510675, −4.19534408676876515339136734135, −4.07128398660108802194326901160, −3.74774372119272103747385329649, −3.46516530021684338870367655111, −3.19345795239035747279391207837, −3.03434076057054052672511575779, −2.74490205880020364820809711640, −2.66718663805887832113118026964, −2.16657055686435846030792150781, −2.02187884507224915197665346298, −1.56622624131455373802495108463, −0.964348526728723522635345045932, −0.885404469964765299552880114964, −0.47319725146674052952374385750, −0.47230786589323883550553166452,
0.47230786589323883550553166452, 0.47319725146674052952374385750, 0.885404469964765299552880114964, 0.964348526728723522635345045932, 1.56622624131455373802495108463, 2.02187884507224915197665346298, 2.16657055686435846030792150781, 2.66718663805887832113118026964, 2.74490205880020364820809711640, 3.03434076057054052672511575779, 3.19345795239035747279391207837, 3.46516530021684338870367655111, 3.74774372119272103747385329649, 4.07128398660108802194326901160, 4.19534408676876515339136734135, 4.30398010880638423186059510675, 5.07141341490606087655331135246, 5.15517217941081665173772338481, 5.19004240919297740875374580913, 5.41725201256928369940354094521, 5.53878511326230623713156313364, 5.59042594194994772843835997551, 6.35450128789117556701624034911, 6.40768579196158970756535205270, 6.46534334968403968896644982098
