| L(s) = 1 | − 4-s − 2·9-s − 14·11-s − 3·16-s − 4·19-s − 4·29-s + 16·31-s + 2·36-s + 18·41-s + 14·44-s − 8·49-s − 48·59-s + 20·61-s + 3·64-s − 28·71-s + 4·76-s + 48·79-s + 3·81-s − 16·89-s + 28·99-s − 26·101-s − 2·109-s + 4·116-s + 87·121-s − 16·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2/3·9-s − 4.22·11-s − 3/4·16-s − 0.917·19-s − 0.742·29-s + 2.87·31-s + 1/3·36-s + 2.81·41-s + 2.11·44-s − 8/7·49-s − 6.24·59-s + 2.56·61-s + 3/8·64-s − 3.32·71-s + 0.458·76-s + 5.40·79-s + 1/3·81-s − 1.69·89-s + 2.81·99-s − 2.58·101-s − 0.191·109-s + 0.371·116-s + 7.90·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7563406849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7563406849\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_b_a_e |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 46 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_i_a_bu |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.o_ef_wc_dhk |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abs_a_bfq |
| 17 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aq_a_be |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.e_bw_gi_cag |
| 23 | $D_4\times C_2$ | \( 1 - 43 T^{2} + 1176 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_abr_a_btg |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.31.aq_im_acpc_rso |
| 37 | $D_4\times C_2$ | \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abf_a_aeq |
| 41 | $D_{4}$ | \( ( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.as_ib_acus_uwi |
| 43 | $D_4\times C_2$ | \( 1 - 91 T^{2} + 5424 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_adn_a_iaq |
| 47 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 1074 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_i_a_abpi |
| 53 | $D_4\times C_2$ | \( 1 - 143 T^{2} + 10216 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_afn_a_pcy |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.59.bw_bqi_wuq_iekc |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.au_nw_afqy_ccbm |
| 67 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 8766 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_bo_a_mze |
| 71 | $D_{4}$ | \( ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bc_uy_kdw_dwws |
| 73 | $D_4\times C_2$ | \( 1 - 175 T^{2} + 15216 T^{4} - 175 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_agt_a_wng |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) | 4.79.abw_btk_abbbk_lcag |
| 83 | $D_4\times C_2$ | \( 1 - 323 T^{2} + 39856 T^{4} - 323 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aml_a_cgyy |
| 89 | $C_4$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.q_me_fci_cluc |
| 97 | $D_4\times C_2$ | \( 1 - 311 T^{2} + 42960 T^{4} - 311 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_alz_a_cloi |
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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
−6.38886708310185818096181551809, −6.37172819632449471808324337488, −5.78147631065495380273705619340, −5.78055829869145500749852374058, −5.65421265092610980269431655960, −5.37289686107096121756035362722, −5.31353517840059737139040302142, −4.83892974456055030360231832003, −4.67115838437079923079693267772, −4.55792882557521425906612668622, −4.53099212439768909408156804900, −4.13209492771647556333732171127, −4.10453349609749920308058363473, −3.39826119013099505134550932173, −3.13864945625753608646767699428, −3.10094996730921783943123629256, −2.78158736611961523639792596892, −2.76312201184365734867181228665, −2.39994278312980442165263612156, −2.09133606814422816123499123628, −2.00275038207825084095357549378, −1.51057248231330080323362777799, −0.938075068546832454543000935030, −0.41124233683428177989139806321, −0.30466270150442596889614971048,
0.30466270150442596889614971048, 0.41124233683428177989139806321, 0.938075068546832454543000935030, 1.51057248231330080323362777799, 2.00275038207825084095357549378, 2.09133606814422816123499123628, 2.39994278312980442165263612156, 2.76312201184365734867181228665, 2.78158736611961523639792596892, 3.10094996730921783943123629256, 3.13864945625753608646767699428, 3.39826119013099505134550932173, 4.10453349609749920308058363473, 4.13209492771647556333732171127, 4.53099212439768909408156804900, 4.55792882557521425906612668622, 4.67115838437079923079693267772, 4.83892974456055030360231832003, 5.31353517840059737139040302142, 5.37289686107096121756035362722, 5.65421265092610980269431655960, 5.78055829869145500749852374058, 5.78147631065495380273705619340, 6.37172819632449471808324337488, 6.38886708310185818096181551809
