| L(s) = 1 | + 10·9-s + 24·17-s + 14·25-s − 4·49-s − 64·73-s + 57·81-s − 36·89-s + 4·97-s + 60·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 240·153-s + 157-s + 163-s + 167-s − 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 5.82·17-s + 14/5·25-s − 4/7·49-s − 7.49·73-s + 19/3·81-s − 3.81·89-s + 0.406·97-s + 5.64·113-s − 0.181·121-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 + 19.4·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.38·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.974969644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.974969644\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ak_a_br |
| 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ao_a_dv |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_e_a_dy |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.13.a_bs_a_bfq |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.17.ay_ky_adci_pja |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abs_a_buk |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bm_a_ccp |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_au_a_cqo |
| 31 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_aba_a_dcl |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.37.a_adq_a_hih |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_agi_a_pkw |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_abc_a_gvm |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aik_a_bbwt |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.61.a_ajk_a_bhas |
| 67 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_ago_a_xzf |
| 71 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ak_a_oyt |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) | 4.73.cm_csi_bszg_shzq |
| 79 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_lg_a_bxzy |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aka_a_btjy |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) | 4.89.bk_bgk_snw_hzzn |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.97.ae_pe_absy_dhgd |
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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
−7.42254266415010265191862959096, −7.41402375225125403165370161927, −7.05896332798379408584246495872, −6.96433372059328318705061617770, −6.87626491670996143902003019847, −6.17465578056601178892860017026, −6.07357610635941163034858211613, −5.81841766254500365045454647694, −5.77076205704955591849517069649, −5.26026512828773627075860722213, −4.99979586651576116876460589906, −4.92207945894315160618051605306, −4.69992620765822386734543424941, −4.14454708434218519772413874022, −4.11123181537214961994069203673, −3.94651062522826954123576818077, −3.30676161783688637168168975419, −3.12869633241877550927825856119, −3.08937256833990610256027871287, −2.86844242770702398423807693370, −2.06782528370611665527851561377, −1.45406621668229034315060462451, −1.30316284317330640005001058506, −1.29207891662806517152795019714, −0.891731207198257283855288652507,
0.891731207198257283855288652507, 1.29207891662806517152795019714, 1.30316284317330640005001058506, 1.45406621668229034315060462451, 2.06782528370611665527851561377, 2.86844242770702398423807693370, 3.08937256833990610256027871287, 3.12869633241877550927825856119, 3.30676161783688637168168975419, 3.94651062522826954123576818077, 4.11123181537214961994069203673, 4.14454708434218519772413874022, 4.69992620765822386734543424941, 4.92207945894315160618051605306, 4.99979586651576116876460589906, 5.26026512828773627075860722213, 5.77076205704955591849517069649, 5.81841766254500365045454647694, 6.07357610635941163034858211613, 6.17465578056601178892860017026, 6.87626491670996143902003019847, 6.96433372059328318705061617770, 7.05896332798379408584246495872, 7.41402375225125403165370161927, 7.42254266415010265191862959096
