| L(s) = 1 | + 2·3-s + 4·7-s − 4·9-s − 4·17-s − 4·19-s + 8·21-s − 3·23-s − 13·27-s − 4·29-s − 10·31-s + 10·37-s − 4·41-s − 24·43-s + 16·47-s − 49-s − 8·51-s − 10·53-s − 8·57-s + 11·59-s − 4·61-s − 16·63-s + 12·67-s − 6·69-s − 4·71-s − 4·73-s + 4·79-s + 4·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s − 4/3·9-s − 0.970·17-s − 0.917·19-s + 1.74·21-s − 0.625·23-s − 2.50·27-s − 0.742·29-s − 1.79·31-s + 1.64·37-s − 0.624·41-s − 3.65·43-s + 2.33·47-s − 1/7·49-s − 1.12·51-s − 1.37·53-s − 1.05·57-s + 1.43·59-s − 0.512·61-s − 2.01·63-s + 1.46·67-s − 0.722·69-s − 0.474·71-s − 0.468·73-s + 0.450·79-s + 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_i_al |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 48 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_r_abw |
| 11 | $A_4\times C_2$ | \( 1 + 5 T^{2} - 56 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.11.a_f_ace |
| 13 | $A_4\times C_2$ | \( 1 + 32 T^{2} + 7 T^{3} + 32 p T^{4} + p^{3} T^{6} \) | 3.13.a_bg_h |
| 17 | $A_4\times C_2$ | \( 1 + 4 T + 47 T^{2} + 128 T^{3} + 47 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bv_ey |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 88 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_z_dk |
| 29 | $A_4\times C_2$ | \( 1 + 4 T + 48 T^{2} + 63 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_bw_cl |
| 31 | $A_4\times C_2$ | \( 1 + 10 T + 110 T^{2} + 579 T^{3} + 110 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.k_eg_wh |
| 37 | $A_4\times C_2$ | \( 1 - 10 T + 79 T^{2} - 412 T^{3} + 79 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ak_db_apw |
| 41 | $A_4\times C_2$ | \( 1 + 4 T + 84 T^{2} + 9 p T^{3} + 84 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_dg_of |
| 43 | $A_4\times C_2$ | \( 1 + 24 T + 293 T^{2} + 2296 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.y_lh_dki |
| 47 | $A_4\times C_2$ | \( 1 - 16 T + 210 T^{2} - 1587 T^{3} + 210 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.aq_ic_acjb |
| 53 | $A_4\times C_2$ | \( 1 + 10 T + 155 T^{2} + 956 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_fz_bku |
| 59 | $A_4\times C_2$ | \( 1 - 11 T + 152 T^{2} - 919 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.al_fw_abjj |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 67 T^{2} + 592 T^{3} + 67 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.e_cp_wu |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 137 T^{2} - 776 T^{3} + 137 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_fh_abdw |
| 71 | $A_4\times C_2$ | \( 1 + 4 T + 62 T^{2} + 35 T^{3} + 62 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_ck_bj |
| 73 | $A_4\times C_2$ | \( 1 + 4 T + 68 T^{2} + 933 T^{3} + 68 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.e_cq_bjx |
| 79 | $A_4\times C_2$ | \( 1 - 4 T + 121 T^{2} - 64 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_er_acm |
| 83 | $A_4\times C_2$ | \( 1 + 8 T + 9 T^{2} - 528 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.i_j_aui |
| 89 | $A_4\times C_2$ | \( 1 + 20 T + 335 T^{2} + 3568 T^{3} + 335 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_mx_fhg |
| 97 | $A_4\times C_2$ | \( 1 + 38 T + 763 T^{2} + 9284 T^{3} + 763 p T^{4} + 38 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bm_bdj_ntc |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.15796779806279862841327625393, −6.99224047532709632148735308729, −6.78767617628797621066407522680, −6.58181693530708783904598924828, −6.07876706490529199376710919043, −5.95728675521972781016836944493, −5.85406550217463827427278149096, −5.50141578734832229142409355013, −5.36436252957850652901608631430, −5.09967501749152456237686005249, −4.69250903125890954697082620683, −4.57029260845146609932108759799, −4.50178053104816807683561095678, −3.95512181693376616919965268699, −3.69707693699368556722118300157, −3.68654243548973543276474550291, −3.30370400923922953576127806201, −3.02475471048627291987481241491, −2.72912270933115836883252297443, −2.30289083797555624335993085034, −2.25836277391358607303534025275, −2.08978424644857753285583488001, −1.69698823188003373273587609638, −1.28484294868244996964887106884, −1.15203804865369486287199810430, 0, 0, 0,
1.15203804865369486287199810430, 1.28484294868244996964887106884, 1.69698823188003373273587609638, 2.08978424644857753285583488001, 2.25836277391358607303534025275, 2.30289083797555624335993085034, 2.72912270933115836883252297443, 3.02475471048627291987481241491, 3.30370400923922953576127806201, 3.68654243548973543276474550291, 3.69707693699368556722118300157, 3.95512181693376616919965268699, 4.50178053104816807683561095678, 4.57029260845146609932108759799, 4.69250903125890954697082620683, 5.09967501749152456237686005249, 5.36436252957850652901608631430, 5.50141578734832229142409355013, 5.85406550217463827427278149096, 5.95728675521972781016836944493, 6.07876706490529199376710919043, 6.58181693530708783904598924828, 6.78767617628797621066407522680, 6.99224047532709632148735308729, 7.15796779806279862841327625393
