| L(s) = 1 | + 3·7-s − 6·11-s + 2·13-s − 3·19-s − 6·23-s − 25-s − 2·29-s + 4·31-s + 6·37-s + 12·41-s + 8·43-s − 4·47-s + 6·49-s − 6·53-s + 18·61-s + 6·71-s + 10·73-s − 18·77-s + 20·79-s + 4·83-s + 12·89-s + 6·91-s − 6·97-s − 12·101-s + 24·103-s + 2·107-s − 6·109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.80·11-s + 0.554·13-s − 0.688·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.986·37-s + 1.87·41-s + 1.21·43-s − 0.583·47-s + 6/7·49-s − 0.824·53-s + 2.30·61-s + 0.712·71-s + 1.17·73-s − 2.05·77-s + 2.25·79-s + 0.439·83-s + 1.27·89-s + 0.628·91-s − 0.609·97-s − 1.19·101-s + 2.36·103-s + 0.193·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.324607713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.324607713\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 + T^{2} - 16 T^{3} + p T^{4} + p^{3} T^{6} \) | 3.5.a_b_aq |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.11.g_bt_fk |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_p_abk |
| 17 | $S_4\times C_2$ | \( 1 + 37 T^{2} - 16 T^{3} + 37 p T^{4} + p^{3} T^{6} \) | 3.17.a_bl_aq |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.23.g_dd_ky |
| 29 | $D_{6}$ | \( 1 + 2 T - 11 T^{2} - 296 T^{3} - 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_al_alk |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 216 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_cv_aii |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 67 T^{2} - 468 T^{3} + 67 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_cp_asa |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.41.am_gp_aboi |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 480 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ai_dh_asm |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} - 272 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.e_p_akm |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 157 T^{2} + 632 T^{3} + 157 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_gb_yi |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.59.a_gv_a |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 235 T^{2} - 2204 T^{3} + 235 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.as_jb_adgu |
| 67 | $S_4\times C_2$ | \( 1 + 145 T^{2} - 128 T^{3} + 145 p T^{4} + p^{3} T^{6} \) | 3.67.a_fp_aey |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 211 T^{2} - 848 T^{3} + 211 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ag_id_abgq |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 151 T^{2} - 1164 T^{3} + 151 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_fv_absu |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 245 T^{2} - 2168 T^{3} + 245 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.au_jl_adfk |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 155 T^{2} - 880 T^{3} + 155 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ae_fz_abhw |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.89.am_md_adgq |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.97.g_lr_btc |
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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
−7.47388037568272191904455908028, −7.22457053962280901337642322414, −6.78198260623656034077653943532, −6.60591933293205581966952345133, −6.15378104366534507591352389367, −6.09202771701938751495185714544, −6.07163969672511990257431250265, −5.37356461999106883584944244057, −5.36870529312430896292562431256, −5.21459860890261353705632363201, −4.90073649282574003463116397172, −4.45539645557523814972599415501, −4.43992733834217977085799158121, −3.94205510183702123129668077054, −3.85133339867108035484888162138, −3.67499095781995070956076402195, −2.94396996101143365399959939923, −2.85538130259959517483978223082, −2.61492271612151799254123814915, −2.04936227104208245380896654922, −1.98789198409049578848396519701, −1.88759697978961040943738250722, −1.03180243000369693299706856890, −0.73338856144486033953605719948, −0.46953722464500792643323454824,
0.46953722464500792643323454824, 0.73338856144486033953605719948, 1.03180243000369693299706856890, 1.88759697978961040943738250722, 1.98789198409049578848396519701, 2.04936227104208245380896654922, 2.61492271612151799254123814915, 2.85538130259959517483978223082, 2.94396996101143365399959939923, 3.67499095781995070956076402195, 3.85133339867108035484888162138, 3.94205510183702123129668077054, 4.43992733834217977085799158121, 4.45539645557523814972599415501, 4.90073649282574003463116397172, 5.21459860890261353705632363201, 5.36870529312430896292562431256, 5.37356461999106883584944244057, 6.07163969672511990257431250265, 6.09202771701938751495185714544, 6.15378104366534507591352389367, 6.60591933293205581966952345133, 6.78198260623656034077653943532, 7.22457053962280901337642322414, 7.47388037568272191904455908028
