| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 3·5-s + 6·6-s + 4·7-s − 4·8-s + 6·9-s − 6·10-s − 5·11-s − 9·12-s + 2·13-s − 8·14-s − 9·15-s + 3·16-s − 12·18-s − 5·19-s + 9·20-s − 12·21-s + 10·22-s + 4·23-s + 12·24-s + 6·25-s − 4·26-s − 10·27-s + 12·28-s − 29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 1.34·5-s + 2.44·6-s + 1.51·7-s − 1.41·8-s + 2·9-s − 1.89·10-s − 1.50·11-s − 2.59·12-s + 0.554·13-s − 2.13·14-s − 2.32·15-s + 3/4·16-s − 2.82·18-s − 1.14·19-s + 2.01·20-s − 2.61·21-s + 2.13·22-s + 0.834·23-s + 2.44·24-s + 6/5·25-s − 0.784·26-s − 1.92·27-s + 2.26·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040917526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040917526\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 17 | | \( 1 \) | |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_b_a |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 18 T^{2} - 46 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_s_abu |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 46 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_r_bu |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 32 T^{2} - 48 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bg_abw |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 32 T^{2} + 129 T^{3} + 32 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.f_bg_ez |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 152 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_bp_afw |
| 29 | $S_4\times C_2$ | \( 1 + T + 79 T^{2} + 54 T^{3} + 79 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.29.b_db_cc |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 86 T^{2} - 396 T^{3} + 86 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ao_di_apg |
| 37 | $S_4\times C_2$ | \( 1 - 28 T + 364 T^{2} - 2806 T^{3} + 364 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.abc_oa_aedy |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 83 T^{2} - 262 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ad_df_akc |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 178 T^{3} + 6 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_g_gw |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 388 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_cn_oy |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 131 T^{2} - 628 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fb_aye |
| 59 | $S_4\times C_2$ | \( 1 + 13 T + 225 T^{2} + 1578 T^{3} + 225 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.n_ir_cis |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 206 T^{2} - 1533 T^{3} + 206 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.an_hy_acgz |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 86 T^{2} + 318 T^{3} + 86 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.e_di_mg |
| 71 | $S_4\times C_2$ | \( 1 + T + 85 T^{2} + 102 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.71.b_dh_dy |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) | 3.73.as_mp_aefk |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 229 T^{2} + 1090 T^{3} + 229 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.h_iv_bpy |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 193 T^{2} + 2340 T^{3} + 193 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_hl_dma |
| 89 | $S_4\times C_2$ | \( 1 - 13 T + 251 T^{2} - 2318 T^{3} + 251 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.an_jr_adle |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 176 T^{2} + 558 T^{3} + 176 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_gu_vm |
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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
−7.34050683945560308730860785357, −7.30492657107267992383172544825, −6.94394698556595719403349808648, −6.47702590788842527043385831837, −6.42071549987793210047917974105, −6.16581795711793354873327711019, −6.09806824813318982155615946332, −5.84434374308771210505995173852, −5.35842581015605115007068419602, −5.33951547701670048707787279529, −4.93722783843114435740587055278, −4.76632423072307864811005207231, −4.51478991090187915699017860678, −4.20013782122776762196629310835, −4.14771315800554221310793740447, −3.23858664826181477866959291232, −3.11369487669156647740332899814, −2.69801013999113107011162752657, −2.31737747900749839391239920487, −2.16562834743516299153812461097, −1.89293989750943542597488918794, −1.42742962301492024975671011793, −1.07076484976584758364397885026, −0.68176478160211260666513267810, −0.53975474701238450842430766960,
0.53975474701238450842430766960, 0.68176478160211260666513267810, 1.07076484976584758364397885026, 1.42742962301492024975671011793, 1.89293989750943542597488918794, 2.16562834743516299153812461097, 2.31737747900749839391239920487, 2.69801013999113107011162752657, 3.11369487669156647740332899814, 3.23858664826181477866959291232, 4.14771315800554221310793740447, 4.20013782122776762196629310835, 4.51478991090187915699017860678, 4.76632423072307864811005207231, 4.93722783843114435740587055278, 5.33951547701670048707787279529, 5.35842581015605115007068419602, 5.84434374308771210505995173852, 6.09806824813318982155615946332, 6.16581795711793354873327711019, 6.42071549987793210047917974105, 6.47702590788842527043385831837, 6.94394698556595719403349808648, 7.30492657107267992383172544825, 7.34050683945560308730860785357
