| L(s) = 1 | − 3·2-s + 6·4-s + 2·7-s − 10·8-s − 2·9-s + 6·11-s − 3·13-s − 6·14-s + 15·16-s − 4·17-s + 6·18-s + 2·19-s − 18·22-s − 6·23-s + 9·26-s + 4·27-s + 12·28-s − 2·29-s + 12·31-s − 21·32-s + 12·34-s − 12·36-s + 8·37-s − 6·38-s + 2·41-s + 12·43-s + 36·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 0.755·7-s − 3.53·8-s − 2/3·9-s + 1.80·11-s − 0.832·13-s − 1.60·14-s + 15/4·16-s − 0.970·17-s + 1.41·18-s + 0.458·19-s − 3.83·22-s − 1.25·23-s + 1.76·26-s + 0.769·27-s + 2.26·28-s − 0.371·29-s + 2.15·31-s − 3.71·32-s + 2.05·34-s − 2·36-s + 1.31·37-s − 0.973·38-s + 0.312·41-s + 1.82·43-s + 5.42·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.069782334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.069782334\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T^{2} - 4 T^{3} + 2 p T^{4} + p^{3} T^{6} \) | 3.3.a_c_ae |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 6 T^{2} - 8 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_g_ai |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.11.ag_bt_afk |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 52 T^{3} + 8 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_i_aca |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 116 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ac_n_aem |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 260 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cb_ka |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_br_ga |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 113 T^{2} - 664 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.am_ej_azo |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 112 T^{2} - 590 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_ei_aws |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_br_ga |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 736 T^{3} + 114 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.am_ek_abci |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 158 T^{2} - 932 T^{3} + 158 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_gc_abjw |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 143 T^{2} + 620 T^{3} + 143 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_fn_xw |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 133 T^{2} - 276 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ac_fd_akq |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ak_ff_abwe |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1528 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_in_acgu |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 178 T^{2} + 936 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_gw_bka |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) | 3.73.as_mp_aefk |
| 79 | $S_4\times C_2$ | \( 1 - 28 T + 453 T^{2} - 4744 T^{3} + 453 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.abc_rl_aham |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 289 T^{2} - 2496 T^{3} + 289 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aq_ld_adsa |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 223 T^{2} + 396 T^{3} + 223 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_ip_pg |
| 97 | $S_4\times C_2$ | \( 1 - 26 T + 343 T^{2} - 3452 T^{3} + 343 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.aba_nf_afcu |
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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
−9.349579580210672123101968491880, −9.134731700858641157000767503858, −8.812122768267532144694526647989, −8.765794372097970688535025788944, −8.054128040875269837098849023971, −8.029796984462013709955664340669, −7.85351887663922066212020501642, −7.56621365417059908575204108803, −7.12664628801960118285361409177, −6.62526042386027785038040592172, −6.45569326550960426610846889497, −6.40949053366096565702776035108, −6.02670337933847354491310509302, −5.49205000164514221989237480455, −5.12286219538976488854882212668, −4.73390575704240133134896809097, −4.27776139549447990596891865554, −3.84596591001573898116851026516, −3.63258094491979259447341770518, −2.80715986911783268604616530037, −2.38491429607640717783989306283, −2.34248822110989210361132930450, −1.70576803094784672782775936339, −0.844132271415937913603633628376, −0.818793260995760882584396948218,
0.818793260995760882584396948218, 0.844132271415937913603633628376, 1.70576803094784672782775936339, 2.34248822110989210361132930450, 2.38491429607640717783989306283, 2.80715986911783268604616530037, 3.63258094491979259447341770518, 3.84596591001573898116851026516, 4.27776139549447990596891865554, 4.73390575704240133134896809097, 5.12286219538976488854882212668, 5.49205000164514221989237480455, 6.02670337933847354491310509302, 6.40949053366096565702776035108, 6.45569326550960426610846889497, 6.62526042386027785038040592172, 7.12664628801960118285361409177, 7.56621365417059908575204108803, 7.85351887663922066212020501642, 8.029796984462013709955664340669, 8.054128040875269837098849023971, 8.765794372097970688535025788944, 8.812122768267532144694526647989, 9.134731700858641157000767503858, 9.349579580210672123101968491880
