| L(s) = 1 | + 3·5-s − 9·9-s + 5·11-s + 2·13-s − 11·17-s + 7·19-s − 4·23-s − 4·25-s − 7·29-s − 11·31-s + 37-s − 2·41-s − 5·43-s − 27·45-s − 19·47-s − 49-s + 6·53-s + 15·55-s + 23·59-s + 4·61-s + 6·65-s + 4·67-s − 4·71-s + 8·73-s − 2·79-s + 54·81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 3·9-s + 1.50·11-s + 0.554·13-s − 2.66·17-s + 1.60·19-s − 0.834·23-s − 4/5·25-s − 1.29·29-s − 1.97·31-s + 0.164·37-s − 0.312·41-s − 0.762·43-s − 4.02·45-s − 2.77·47-s − 1/7·49-s + 0.824·53-s + 2.02·55-s + 2.99·59-s + 0.512·61-s + 0.744·65-s + 0.488·67-s − 0.474·71-s + 0.936·73-s − 0.225·79-s + 6·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 151^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272091768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272091768\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 151 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.3.a_j_a |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 13 T^{2} - 27 T^{3} + 13 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_n_abb |
| 7 | $S_4\times C_2$ | \( 1 + T^{2} - 8 T^{3} + p T^{4} + p^{3} T^{6} \) | 3.7.a_b_ai |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 31 T^{2} - 95 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.af_bf_adr |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} + 36 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_h_bk |
| 17 | $S_4\times C_2$ | \( 1 + 11 T + 81 T^{2} + 383 T^{3} + 81 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.l_dd_ot |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 51 T^{2} - 267 T^{3} + 51 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ah_bz_akh |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 208 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_bp_ia |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 81 T^{2} + 407 T^{3} + 81 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.h_dd_pr |
| 31 | $S_4\times C_2$ | \( 1 + 11 T + 101 T^{2} + 559 T^{3} + 101 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.l_dx_vn |
| 37 | $S_4\times C_2$ | \( 1 - T + 49 T^{2} - 237 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ab_bx_ajd |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 59 T^{2} + 236 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_ch_jc |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 77 T^{2} + 481 T^{3} + 77 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_cz_sn |
| 47 | $S_4\times C_2$ | \( 1 + 19 T + 239 T^{2} + 1939 T^{3} + 239 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.t_jf_cwp |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 151 T^{2} - 612 T^{3} + 151 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fv_axo |
| 59 | $S_4\times C_2$ | \( 1 - 23 T + 321 T^{2} - 2863 T^{3} + 321 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ax_mj_aegd |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 51 T^{2} + 304 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ae_bz_ls |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 173 T^{2} - 560 T^{3} + 173 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ae_gr_avo |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 504 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_cb_tk |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 111 T^{2} - 1272 T^{3} + 111 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ai_eh_abwy |
| 79 | $S_4\times C_2$ | \( 1 + 2 T - 23 T^{2} - 780 T^{3} - 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.c_ax_abea |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 1568 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.q_ht_cii |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{3} \) | 3.89.abq_bgx_apdc |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 217 T^{2} - 627 T^{3} + 217 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ad_ij_ayd |
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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
−6.80213158236446206619887480346, −6.38964811640562645208561848646, −6.27416082598586975471292167728, −6.18134813439860435772866853849, −5.75051814617725926250011922822, −5.66244282185232052548374616896, −5.61046495767930824148940239782, −5.20298152443993396232649793981, −5.05849415925182399701703120227, −4.88466557830717569832178716110, −4.35513249010070214233161253403, −4.21244655297991708399128679209, −3.81842370179961153612737104906, −3.49041883518588746529662525166, −3.41225282368917513083677328781, −3.37439438014819108876405610260, −2.93923925208476899831958627441, −2.31237539470184667163163163138, −2.26582534810741652313082166791, −2.11082432236709257248596690702, −1.78269653733886100391216917526, −1.72128086040645833864737764121, −1.01654304140979442217026731576, −0.49219527171272688810830573609, −0.31624688761116022677334347722,
0.31624688761116022677334347722, 0.49219527171272688810830573609, 1.01654304140979442217026731576, 1.72128086040645833864737764121, 1.78269653733886100391216917526, 2.11082432236709257248596690702, 2.26582534810741652313082166791, 2.31237539470184667163163163138, 2.93923925208476899831958627441, 3.37439438014819108876405610260, 3.41225282368917513083677328781, 3.49041883518588746529662525166, 3.81842370179961153612737104906, 4.21244655297991708399128679209, 4.35513249010070214233161253403, 4.88466557830717569832178716110, 5.05849415925182399701703120227, 5.20298152443993396232649793981, 5.61046495767930824148940239782, 5.66244282185232052548374616896, 5.75051814617725926250011922822, 6.18134813439860435772866853849, 6.27416082598586975471292167728, 6.38964811640562645208561848646, 6.80213158236446206619887480346
