| L(s) = 1 | + 4-s + 4·5-s + 6·7-s − 6·9-s + 12·11-s + 12·13-s − 6·17-s + 6·19-s + 4·20-s + 5·25-s + 6·28-s − 4·31-s + 24·35-s − 6·36-s + 2·41-s + 12·44-s − 24·45-s + 8·49-s + 12·52-s − 36·53-s + 48·55-s − 12·59-s − 12·61-s − 36·63-s − 64-s + 48·65-s − 6·67-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s + 2.26·7-s − 2·9-s + 3.61·11-s + 3.32·13-s − 1.45·17-s + 1.37·19-s + 0.894·20-s + 25-s + 1.13·28-s − 0.718·31-s + 4.05·35-s − 36-s + 0.312·41-s + 1.80·44-s − 3.57·45-s + 8/7·49-s + 1.66·52-s − 4.94·53-s + 6.47·55-s − 1.56·59-s − 1.53·61-s − 4.53·63-s − 1/8·64-s + 5.95·65-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.479691449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.479691449\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_g_a_bb |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ag_bc_ads_lf |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.am_do_asa_cry |
| 13 | $D_4\times C_2$ | \( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1611 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.am_de_aps_cjz |
| 17 | $C_2^2$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.g_bx_io_bxk |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_ai_abk_bih |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_aci_a_cpy |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_eg_a_gzb |
| 31 | $D_{4}$ | \( ( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.e_au_cy_dla |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 31 T^{2} + 94 T^{3} - 620 T^{4} + 94 p T^{5} - 31 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ac_abf_dq_axw |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_acq_a_gic |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_afc_a_mny |
| 53 | $D_4\times C_2$ | \( 1 + 36 T + 642 T^{2} + 7560 T^{3} + 64187 T^{4} + 7560 p T^{5} + 642 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.bk_ys_leu_dqyt |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 288 p T^{5} + 2 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.m_c_lc_mcx |
| 61 | $D_4\times C_2$ | \( 1 + 12 T - 11 T^{2} + 396 T^{3} + 11520 T^{4} + 396 p T^{5} - 11 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.m_al_pg_rbc |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 100 T^{2} + 528 T^{3} + 4059 T^{4} + 528 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.g_dw_ui_gad |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 32 T^{2} - 828 T^{3} - 7581 T^{4} - 828 p T^{5} + 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.g_bg_abfw_alfp |
| 73 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 10086 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_dw_a_oxy |
| 79 | $D_4\times C_2$ | \( 1 - 14 T + 64 T^{2} + 364 T^{3} - 4301 T^{4} + 364 p T^{5} + 64 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ao_cm_oa_agjl |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 190 T^{2} - 1704 T^{3} + 17259 T^{4} - 1704 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.am_hi_acno_znv |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 139 T^{2} - 18 T^{3} + 19500 T^{4} - 18 p T^{5} - 139 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.g_afj_as_bcwa |
| 97 | $D_4\times C_2$ | \( 1 - 326 T^{2} + 44955 T^{4} - 326 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_amo_a_conb |
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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
−8.477792462110918306809766267965, −7.913450999327177245951913710069, −7.895672064782375657627623634292, −7.81300621648394385768618414519, −7.25155308099583979509767833446, −6.57529689787088997706179133007, −6.55367934146860195117975878771, −6.45660061565450139267834914596, −6.38372774454026419802330215857, −6.07104127152737529247015358375, −5.69832737446367516151538623705, −5.55479424027320720636749876189, −5.35247627093593017980828345601, −4.81457003188789232725764168001, −4.54315880618533506643770970511, −4.33458277560117063201350108713, −3.97058560409520192777649391883, −3.49082784379341259265791973288, −3.36854791999168132351616504737, −3.00117636607862617056858553965, −2.53040825982049553068896613635, −1.70776531494346732372960374905, −1.57424150231961967945257086400, −1.56430437007422083752390364431, −1.28483020143125582087777913855,
1.28483020143125582087777913855, 1.56430437007422083752390364431, 1.57424150231961967945257086400, 1.70776531494346732372960374905, 2.53040825982049553068896613635, 3.00117636607862617056858553965, 3.36854791999168132351616504737, 3.49082784379341259265791973288, 3.97058560409520192777649391883, 4.33458277560117063201350108713, 4.54315880618533506643770970511, 4.81457003188789232725764168001, 5.35247627093593017980828345601, 5.55479424027320720636749876189, 5.69832737446367516151538623705, 6.07104127152737529247015358375, 6.38372774454026419802330215857, 6.45660061565450139267834914596, 6.55367934146860195117975878771, 6.57529689787088997706179133007, 7.25155308099583979509767833446, 7.81300621648394385768618414519, 7.895672064782375657627623634292, 7.913450999327177245951913710069, 8.477792462110918306809766267965
