| L(s) = 1 | − 6·4-s − 2·9-s + 12·11-s + 19·16-s + 4·19-s − 4·29-s + 24·31-s + 12·36-s − 28·41-s − 72·44-s + 12·49-s − 24·59-s − 24·61-s − 36·64-s + 8·71-s − 24·76-s + 16·79-s + 3·81-s + 36·89-s − 24·99-s + 16·101-s + 24·116-s + 60·121-s − 144·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3·4-s − 2/3·9-s + 3.61·11-s + 19/4·16-s + 0.917·19-s − 0.742·29-s + 4.31·31-s + 2·36-s − 4.37·41-s − 10.8·44-s + 12/7·49-s − 3.12·59-s − 3.07·61-s − 9/2·64-s + 0.949·71-s − 2.75·76-s + 1.80·79-s + 1/3·81-s + 3.81·89-s − 2.41·99-s + 1.59·101-s + 2.22·116-s + 5.45·121-s − 12.9·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304248522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304248522\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 5 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) | 4.2.a_g_a_r |
| 7 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_am_a_ec |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.am_dg_aqe_cjy |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_au_a_he |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abk_a_bis |
| 23 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 730 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ae_a_abcc |
| 29 | $D_{4}$ | \( ( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.e_ae_dw_cwg |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.31.ay_nc_aepc_begs |
| 37 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7066 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_afc_a_klu |
| 41 | $D_{4}$ | \( ( 1 + 14 T + 124 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.bc_rc_gvs_bzmw |
| 43 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8346 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_afk_a_mja |
| 47 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5126 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_adw_a_hpe |
| 53 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 3350 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_adg_a_eyw |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.y_pg_gou_cgxy |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.y_po_gui_cjzy |
| 67 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 1846 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_am_a_cta |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ai_js_acge_bmws |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ado_a_sxi |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.aq_hg_adiu_bncw |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aka_a_btjy |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.abk_bbo_apeq_gjpa |
| 97 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 23754 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_aie_a_bjdq |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.77306864104276983433946453203, −6.54170038930594026168364181781, −6.42035845323397024182214404453, −6.08465483102187199736904327314, −6.00884736493140656197035006263, −5.70881982029232643084603607904, −5.43909063193821190254668619828, −5.17008533026142241298646497908, −4.76076014315921208722936644579, −4.67173632818568316080806737483, −4.66642961651913353231932962902, −4.42609621640050870761065964335, −4.33575912099777727312945573373, −3.61954067693656378523645027394, −3.58859458748282625006603076806, −3.56865178314112894591400262752, −3.46530521443920727581844780315, −2.85540761259727915283466745800, −2.81803512599027198934523148644, −1.93039957994342878797821849001, −1.89395016087929708957627954000, −1.35917482622367926127006304528, −1.00592871596478144731837563201, −0.884996695984924135403672373460, −0.32673721334488739356012600522,
0.32673721334488739356012600522, 0.884996695984924135403672373460, 1.00592871596478144731837563201, 1.35917482622367926127006304528, 1.89395016087929708957627954000, 1.93039957994342878797821849001, 2.81803512599027198934523148644, 2.85540761259727915283466745800, 3.46530521443920727581844780315, 3.56865178314112894591400262752, 3.58859458748282625006603076806, 3.61954067693656378523645027394, 4.33575912099777727312945573373, 4.42609621640050870761065964335, 4.66642961651913353231932962902, 4.67173632818568316080806737483, 4.76076014315921208722936644579, 5.17008533026142241298646497908, 5.43909063193821190254668619828, 5.70881982029232643084603607904, 6.00884736493140656197035006263, 6.08465483102187199736904327314, 6.42035845323397024182214404453, 6.54170038930594026168364181781, 6.77306864104276983433946453203
