| L(s) = 1 | − 10·7-s + 3·9-s − 6·13-s + 4·29-s − 30·37-s + 22·47-s + 43·49-s + 28·61-s − 30·63-s + 20·67-s + 12·73-s + 28·79-s − 7·81-s + 60·91-s + 8·97-s + 4·101-s − 18·117-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.77·7-s + 9-s − 1.66·13-s + 0.742·29-s − 4.93·37-s + 3.20·47-s + 43/7·49-s + 3.58·61-s − 3.77·63-s + 2.44·67-s + 1.40·73-s + 3.15·79-s − 7/9·81-s + 6.28·91-s + 0.812·97-s + 0.398·101-s − 1.66·117-s + 8/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041576519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041576519\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ad_a_q |
| 7 | $D_{4}$ | \( ( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.k_cf_iw_bbc |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ai_a_hi |
| 17 | $D_4\times C_2$ | \( 1 - 55 T^{2} + 1296 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_acd_a_bxw |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ace_a_cqg |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.ae_dk_aky_flm |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adk_a_fpu |
| 37 | $D_{4}$ | \( ( 1 + 15 T + 126 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.be_sj_hgc_caei |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.41.a_abk_a_flu |
| 43 | $D_4\times C_2$ | \( 1 - 51 T^{2} + 3392 T^{4} - 51 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_abz_a_fam |
| 47 | $D_{4}$ | \( ( 1 - 11 T + 120 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.aw_nx_afli_bsrg |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1750 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abs_a_cpi |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abk_a_kug |
| 61 | $D_{4}$ | \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.abc_tk_aixo_ddmc |
| 67 | $D_{4}$ | \( ( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.au_ou_ageu_ckyg |
| 71 | $D_4\times C_2$ | \( 1 - 207 T^{2} + 20756 T^{4} - 207 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ahz_a_besi |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.am_bo_abiq_xog |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abc_we_alds_enrq |
| 83 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ho_a_bipi |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ami_a_cjfi |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ai_kq_acrw_cfle |
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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
−6.47446162178942725914020610640, −6.07173253856647655474903920706, −6.03866152205692567283666116248, −5.68472786711422659121344350904, −5.38407178116572535044873475450, −5.32974057763998084254725424915, −5.10869960519321562795955694688, −4.83266648377018802074424267658, −4.80333515216444536136788011015, −4.40390147654619045259696361818, −3.86497537265539858807967157663, −3.78047439143423070914865656164, −3.73160269945257836760988190966, −3.54894811178749053830617286293, −3.52325038070342955347899796876, −3.03393134292721245810844775875, −2.77791043885869244839235138290, −2.44680091417584673114502559570, −2.42195262904634865033279362704, −2.13903797390252975914510355764, −1.88634788412612798390678070650, −1.26523529779656315235732657733, −0.951815859639121313382064301598, −0.43151567313482041116667762052, −0.31841013166316998929969633249,
0.31841013166316998929969633249, 0.43151567313482041116667762052, 0.951815859639121313382064301598, 1.26523529779656315235732657733, 1.88634788412612798390678070650, 2.13903797390252975914510355764, 2.42195262904634865033279362704, 2.44680091417584673114502559570, 2.77791043885869244839235138290, 3.03393134292721245810844775875, 3.52325038070342955347899796876, 3.54894811178749053830617286293, 3.73160269945257836760988190966, 3.78047439143423070914865656164, 3.86497537265539858807967157663, 4.40390147654619045259696361818, 4.80333515216444536136788011015, 4.83266648377018802074424267658, 5.10869960519321562795955694688, 5.32974057763998084254725424915, 5.38407178116572535044873475450, 5.68472786711422659121344350904, 6.03866152205692567283666116248, 6.07173253856647655474903920706, 6.47446162178942725914020610640
