| L(s) = 1 | + 4·3-s + 10·9-s + 12·13-s − 3·16-s + 12·17-s + 20·27-s + 8·29-s + 48·39-s + 16·47-s − 12·48-s + 48·51-s − 16·71-s + 20·73-s + 16·79-s + 35·81-s + 8·83-s + 32·87-s − 4·97-s + 24·103-s + 32·109-s + 120·117-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 64·141-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 3.32·13-s − 3/4·16-s + 2.91·17-s + 3.84·27-s + 1.48·29-s + 7.68·39-s + 2.33·47-s − 1.73·48-s + 6.72·51-s − 1.89·71-s + 2.34·73-s + 1.80·79-s + 35/9·81-s + 0.878·83-s + 3.43·87-s − 0.406·97-s + 2.36·103-s + 3.06·109-s + 11.0·117-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.38·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(36.12254334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.12254334\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 5 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 + 3 T^{4} + p^{4} T^{8} \) | 4.2.a_a_a_d |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_e_a_jm |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.am_ds_atw_dfq |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.am_ei_azk_euw |
| 19 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 242 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_i_a_aji |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 64 T^{2} + 2062 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_cm_a_dbi |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.29.ai_fk_abca_jog |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 2718 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_cu_a_eao |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 36 T^{2} + 2742 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_bk_a_ebm |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 3006 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_y_a_elq |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 44 T^{2} + 2902 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_bs_a_ehq |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.aq_eu_abvk_qty |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 160 T^{2} + 11518 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ge_a_rba |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 8598 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_ee_a_mss |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 36 T^{2} - 234 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_bk_a_aja |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 156 T^{2} + 14742 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ga_a_vva |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.q_nc_eyq_cenu |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.au_qq_ahbw_dadi |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.aq_js_aecm_bumw |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ai_me_acts_cfpm |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 104 T^{2} + 16926 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ea_a_zba |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.e_fo_zs_bkgc |
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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
−6.03635222696970332372853711043, −6.02944658786660036116118225265, −5.62669559127054153981593078142, −5.53173112175513150322179648760, −5.27019487781283009607866357951, −4.88914286833448265090545321858, −4.67282856612736425552398964030, −4.61314536961020508125399852650, −4.41714035857214583227168475273, −3.96713010882329172185987571512, −3.80454540298679071092539854927, −3.66045409778749543469123830850, −3.57778720919361699044159201686, −3.30126774344454311097563817123, −3.29146938311291220710420616059, −2.99237757793486207088662345017, −2.53121220766337814828883317375, −2.50263008206974355980968714904, −2.26853070607354231097130313648, −1.88981751599083972301318138514, −1.60111497903188291416680904769, −1.35054764093303449612626188804, −0.988862380705954479716006024851, −0.939803166626951861067288946440, −0.66319198782652543156763378256,
0.66319198782652543156763378256, 0.939803166626951861067288946440, 0.988862380705954479716006024851, 1.35054764093303449612626188804, 1.60111497903188291416680904769, 1.88981751599083972301318138514, 2.26853070607354231097130313648, 2.50263008206974355980968714904, 2.53121220766337814828883317375, 2.99237757793486207088662345017, 3.29146938311291220710420616059, 3.30126774344454311097563817123, 3.57778720919361699044159201686, 3.66045409778749543469123830850, 3.80454540298679071092539854927, 3.96713010882329172185987571512, 4.41714035857214583227168475273, 4.61314536961020508125399852650, 4.67282856612736425552398964030, 4.88914286833448265090545321858, 5.27019487781283009607866357951, 5.53173112175513150322179648760, 5.62669559127054153981593078142, 6.02944658786660036116118225265, 6.03635222696970332372853711043
