| L(s) = 1 | + 4·7-s − 12·13-s − 4·19-s − 6·25-s − 20·31-s − 24·37-s + 4·43-s + 10·49-s − 8·61-s + 8·67-s − 32·73-s − 16·79-s − 48·91-s − 16·97-s − 16·103-s − 8·109-s − 20·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 3.32·13-s − 0.917·19-s − 6/5·25-s − 3.59·31-s − 3.94·37-s + 0.609·43-s + 10/7·49-s − 1.02·61-s + 0.977·67-s − 3.74·73-s − 1.80·79-s − 5.03·91-s − 1.62·97-s − 1.57·103-s − 0.766·109-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 + 6 T^{2} + 14 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_g_a_o |
| 11 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 262 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_u_a_kc |
| 13 | $D_{4}$ | \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.m_ds_tw_dfq |
| 17 | $D_4\times C_2$ | \( 1 + 62 T^{2} + 1534 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ck_a_cha |
| 23 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 662 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bk_a_zm |
| 29 | $D_4\times C_2$ | \( 1 + 30 T^{2} + 1502 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_be_a_cfu |
| 31 | $D_{4}$ | \( ( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.u_ke_diy_vza |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.37.y_oa_ffs_blso |
| 41 | $D_4\times C_2$ | \( 1 + 140 T^{2} + 8182 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_fk_a_mcs |
| 43 | $D_{4}$ | \( ( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ae_acu_au_ida |
| 47 | $D_4\times C_2$ | \( 1 + 54 T^{2} + 4022 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_cc_a_fys |
| 53 | $D_4\times C_2$ | \( 1 + 86 T^{2} + 6862 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_di_a_kdy |
| 59 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_adg_a_mxq |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.i_ee_bgy_ras |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ai_js_acey_blbi |
| 71 | $D_4\times C_2$ | \( 1 + 190 T^{2} + 16902 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_hi_a_zac |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.bg_ym_mls_euly |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.q_js_ecm_bumw |
| 83 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 774 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ac_a_bdu |
| 89 | $D_4\times C_2$ | \( 1 + 332 T^{2} + 43318 T^{4} + 332 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_mu_a_cmcc |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.q_eu_dae_bvny |
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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
−6.33869442257665031444111115492, −5.67359951482710780346531902426, −5.65456999533785586061157805079, −5.60952661808464080705745450248, −5.55918971597301619302396750652, −5.21615286885215708657220524400, −5.10838008247957123906965815884, −4.95308963225724869800064640146, −4.72067390786763167319210219407, −4.40755703001939428843051762000, −4.27362038097360279454761887814, −4.17851361570849248701947861362, −4.01109552123616620893476015508, −3.52814031667448356876117648730, −3.51801928325114189344954398220, −3.26885426278856319354476109589, −3.00803301495261763640919971152, −2.50861041805258019857604858802, −2.45384891904683586832325859875, −2.28201215457650067072952343465, −2.18296533398540651323550229840, −1.71173362330190886427672613267, −1.57463874677550758457091818522, −1.38710818612479689931326629165, −1.20905818622487886292865292554, 0, 0, 0, 0,
1.20905818622487886292865292554, 1.38710818612479689931326629165, 1.57463874677550758457091818522, 1.71173362330190886427672613267, 2.18296533398540651323550229840, 2.28201215457650067072952343465, 2.45384891904683586832325859875, 2.50861041805258019857604858802, 3.00803301495261763640919971152, 3.26885426278856319354476109589, 3.51801928325114189344954398220, 3.52814031667448356876117648730, 4.01109552123616620893476015508, 4.17851361570849248701947861362, 4.27362038097360279454761887814, 4.40755703001939428843051762000, 4.72067390786763167319210219407, 4.95308963225724869800064640146, 5.10838008247957123906965815884, 5.21615286885215708657220524400, 5.55918971597301619302396750652, 5.60952661808464080705745450248, 5.65456999533785586061157805079, 5.67359951482710780346531902426, 6.33869442257665031444111115492
