| L(s) = 1 | + 2·3-s − 9-s + 6·13-s + 6·17-s − 4·23-s − 2·27-s − 4·29-s + 12·39-s − 30·43-s + 7·49-s + 12·51-s − 16·53-s + 28·61-s − 8·69-s − 28·79-s − 7·81-s − 8·87-s + 4·101-s + 4·103-s + 16·107-s − 40·113-s − 6·117-s + 8·121-s + 127-s − 60·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s + 1.66·13-s + 1.45·17-s − 0.834·23-s − 0.384·27-s − 0.742·29-s + 1.92·39-s − 4.57·43-s + 49-s + 1.68·51-s − 2.19·53-s + 3.58·61-s − 0.963·69-s − 3.15·79-s − 7/9·81-s − 0.857·87-s + 0.398·101-s + 0.394·103-s + 1.54·107-s − 3.76·113-s − 0.554·117-s + 8/11·121-s + 0.0887·127-s − 5.28·129-s + 0.0873·131-s + 0.0854·137-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
| \(L(1)\) |
\(\approx\) |
\(1.316132537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316132537\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ac_f_ak_bc |
| 7 | $D_4\times C_2$ | \( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.7.a_ah_a_e |
| 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 | $D_{4}$ | \( ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ag_cv_ali_cvk |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.e_cm_ie_dek |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.e_dk_ky_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 | $D_4\times C_2$ | \( 1 - 27 T^{2} + 1964 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abb_a_cxo |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.41.a_abk_a_flu |
| 43 | $D_{4}$ | \( ( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.be_th_iaw_ckgy |
| 47 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 7444 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_aep_a_lai |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.q_gq_cnw_wre |
| 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 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15742 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ahc_a_xhm |
| 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 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_bg_a_hzy |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.bc_we_lds_enrq |
| 83 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aho_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 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 32614 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ajk_a_bwgk |
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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
−6.16128372619080169982688321309, −6.04780709771640764276646226447, −6.02964658932082503354915958994, −5.67263719302833352797810765286, −5.43417612974762829579256200012, −5.30182611021372077828834450083, −5.06380757559388548764527076275, −5.00799411240086536245259401857, −4.57091073299218846933134771886, −4.29048401679741817827782822161, −4.11414882386725882591062481499, −3.84476389002700642242521618400, −3.73020263924308325077919449092, −3.37900841338489458886111696353, −3.19816366042449051823284278268, −3.15833751194880400479225111080, −2.99026151245820897797831999850, −2.48073206435563877992571676780, −2.38460844853698997899480123569, −1.91936350238107778707698841273, −1.74904655819523325533417803644, −1.43631919418231833647400606735, −1.23391290604686250562501914029, −0.806697978422680555379948606074, −0.15596101721368564172118997332,
0.15596101721368564172118997332, 0.806697978422680555379948606074, 1.23391290604686250562501914029, 1.43631919418231833647400606735, 1.74904655819523325533417803644, 1.91936350238107778707698841273, 2.38460844853698997899480123569, 2.48073206435563877992571676780, 2.99026151245820897797831999850, 3.15833751194880400479225111080, 3.19816366042449051823284278268, 3.37900841338489458886111696353, 3.73020263924308325077919449092, 3.84476389002700642242521618400, 4.11414882386725882591062481499, 4.29048401679741817827782822161, 4.57091073299218846933134771886, 5.00799411240086536245259401857, 5.06380757559388548764527076275, 5.30182611021372077828834450083, 5.43417612974762829579256200012, 5.67263719302833352797810765286, 6.02964658932082503354915958994, 6.04780709771640764276646226447, 6.16128372619080169982688321309
