| L(s) = 1 | + 3·3-s + 4·5-s + 3·7-s + 6·9-s + 6·13-s + 12·15-s + 8·17-s − 3·19-s + 9·21-s + 4·23-s + 25-s + 10·27-s + 20·29-s − 8·31-s + 12·35-s − 10·37-s + 18·39-s + 14·41-s + 8·43-s + 24·45-s − 6·47-s + 6·49-s + 24·51-s + 4·53-s − 9·57-s − 12·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.78·5-s + 1.13·7-s + 2·9-s + 1.66·13-s + 3.09·15-s + 1.94·17-s − 0.688·19-s + 1.96·21-s + 0.834·23-s + 1/5·25-s + 1.92·27-s + 3.71·29-s − 1.43·31-s + 2.02·35-s − 1.64·37-s + 2.88·39-s + 2.18·41-s + 1.21·43-s + 3.57·45-s − 0.875·47-s + 6/7·49-s + 3.36·51-s + 0.549·53-s − 1.19·57-s − 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(35.11473020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.11473020\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 36 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ae_p_abk |
| 11 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 16 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.11.a_r_q |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_bj_afs |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 276 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ai_cp_akq |
| 23 | $S_4\times C_2$ | \( 1 - 4 T - 11 T^{2} + 216 T^{3} - 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_al_ii |
| 29 | $S_4\times C_2$ | \( 1 - 20 T + 211 T^{2} - 1404 T^{3} + 211 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.au_id_acca |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 192 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_cb_hk |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 91 T^{2} + 604 T^{3} + 91 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_dn_xg |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 135 T^{2} - 852 T^{3} + 135 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ao_ff_abgu |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 384 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ai_dl_aou |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 640 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_cb_yq |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 131 T^{2} - 308 T^{3} + 131 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ae_fb_alw |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.59.m_ir_cey |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_hn_achw |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 169 T^{2} - 568 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ae_gn_avw |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 121 T^{2} + 552 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.c_er_vg |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 231 T^{2} + 1420 T^{3} + 231 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.k_ix_ccq |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 173 T^{2} - 1248 T^{3} + 173 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aq_gr_abwa |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 161 T^{2} - 1360 T^{3} + 161 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ao_gf_acai |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 279 T^{2} - 2196 T^{3} + 279 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ao_kt_adgm |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 111 T^{2} - 76 T^{3} + 111 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ak_eh_acy |
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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
−7.09027022190393848759636945537, −6.78872950666676014091656489771, −6.62341271809709185585294669670, −6.51292585095381042001414098581, −6.02150693369943066633347654964, −5.93065646049313798679030802948, −5.77094025347754099903101785105, −5.32090616340132813692121499600, −5.19265910463661343380779104762, −5.07915142679677027739972323430, −4.48404157408969383923504103799, −4.41066763954544464457950147737, −4.18591296446556273557616512701, −3.78315036879076249725221220041, −3.45744266942920946511195242267, −3.37323648007305583390434245772, −3.02182689713219053774291796924, −2.63129922227829458749006221291, −2.56808045076113444393180496348, −1.96553711576128718328191987642, −1.85676027230636579894976402182, −1.80848476527898344934846620709, −1.13614286645630935641275243233, −0.942940875774929405817172802449, −0.850392505381359108120248888581,
0.850392505381359108120248888581, 0.942940875774929405817172802449, 1.13614286645630935641275243233, 1.80848476527898344934846620709, 1.85676027230636579894976402182, 1.96553711576128718328191987642, 2.56808045076113444393180496348, 2.63129922227829458749006221291, 3.02182689713219053774291796924, 3.37323648007305583390434245772, 3.45744266942920946511195242267, 3.78315036879076249725221220041, 4.18591296446556273557616512701, 4.41066763954544464457950147737, 4.48404157408969383923504103799, 5.07915142679677027739972323430, 5.19265910463661343380779104762, 5.32090616340132813692121499600, 5.77094025347754099903101785105, 5.93065646049313798679030802948, 6.02150693369943066633347654964, 6.51292585095381042001414098581, 6.62341271809709185585294669670, 6.78872950666676014091656489771, 7.09027022190393848759636945537
