| L(s) = 1 | + 4·3-s + 6·9-s − 8·11-s − 8·13-s + 8·23-s + 8·25-s − 4·27-s − 32·33-s + 8·37-s − 32·39-s − 8·47-s + 20·49-s − 16·59-s − 40·61-s + 32·69-s + 16·71-s + 32·73-s + 32·75-s − 37·81-s − 56·83-s + 8·97-s − 48·99-s − 16·107-s + 56·109-s + 32·111-s − 48·117-s − 4·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 2.41·11-s − 2.21·13-s + 1.66·23-s + 8/5·25-s − 0.769·27-s − 5.57·33-s + 1.31·37-s − 5.12·39-s − 1.16·47-s + 20/7·49-s − 2.08·59-s − 5.12·61-s + 3.85·69-s + 1.89·71-s + 3.74·73-s + 3.69·75-s − 4.11·81-s − 6.14·83-s + 0.812·97-s − 4.82·99-s − 1.54·107-s + 5.36·109-s + 3.03·111-s − 4.43·117-s − 0.363·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109254105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109254105\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ai_a_bi |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_au_a_hq |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.11.i_cq_lk_bww |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.i_bo_hs_bio |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abk_a_bis |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_dc_aqy_eek |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aca_a_dms |
| 31 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 1186 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_aq_a_btq |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 76 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ai_gm_abiu_oja |
| 41 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 2246 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_am_a_dik |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_acy_a_epa |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.i_gu_bnc_sfy |
| 53 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_acq_a_gzu |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.q_ki_dvc_bkwk |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bo_bfw_qhc_fuyo |
| 67 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10774 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_aeu_a_pyk |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.aq_oa_afga_cjqs |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.73.abg_baa_anki_fghq |
| 79 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 7454 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_aq_a_alas |
| 83 | $D_{4}$ | \( ( 1 + 28 T + 354 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ce_cfk_bkfg_pihi |
| 89 | $D_4\times C_2$ | \( 1 - 84 T^{2} - 826 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_adg_a_abfu |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 190 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ai_pg_adki_dhvq |
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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
−6.77348242055046617063871531518, −6.42248767205149374777135795629, −6.28852717532710414088628515408, −5.75615561867657073103177652867, −5.53816451826193688222012136146, −5.52568295532871471796108100808, −5.49804926684948274066847941463, −4.87835333745001743544186931856, −4.74990180939572452933100904072, −4.63153406960399344340168480282, −4.59179412555006535692887462497, −4.23570529015862044576134532851, −3.81574840887749066164950112687, −3.56997830354932363334917380337, −3.21037253056851675967392048469, −2.98202447361615677465504991885, −2.88918177903464592404922988485, −2.78791783023052873768517410260, −2.59329427400690515087099469195, −2.26774197474226934344118460980, −2.11682720905723595589367641501, −1.60395884514799806161910639720, −1.41272252099293075151564846440, −0.67582192778320853721807738689, −0.29415040723603553242020107549,
0.29415040723603553242020107549, 0.67582192778320853721807738689, 1.41272252099293075151564846440, 1.60395884514799806161910639720, 2.11682720905723595589367641501, 2.26774197474226934344118460980, 2.59329427400690515087099469195, 2.78791783023052873768517410260, 2.88918177903464592404922988485, 2.98202447361615677465504991885, 3.21037253056851675967392048469, 3.56997830354932363334917380337, 3.81574840887749066164950112687, 4.23570529015862044576134532851, 4.59179412555006535692887462497, 4.63153406960399344340168480282, 4.74990180939572452933100904072, 4.87835333745001743544186931856, 5.49804926684948274066847941463, 5.52568295532871471796108100808, 5.53816451826193688222012136146, 5.75615561867657073103177652867, 6.28852717532710414088628515408, 6.42248767205149374777135795629, 6.77348242055046617063871531518
