| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s − 7-s − 10·8-s − 9·10-s − 11·11-s + 3·14-s + 15·16-s − 17-s + 18·20-s + 33·22-s + 2·23-s + 6·25-s − 6·28-s + 5·29-s + 3·31-s − 21·32-s + 3·34-s − 3·35-s − 3·37-s − 30·40-s + 9·41-s − 11·43-s − 66·44-s − 6·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s − 0.377·7-s − 3.53·8-s − 2.84·10-s − 3.31·11-s + 0.801·14-s + 15/4·16-s − 0.242·17-s + 4.02·20-s + 7.03·22-s + 0.417·23-s + 6/5·25-s − 1.13·28-s + 0.928·29-s + 0.538·31-s − 3.71·32-s + 0.514·34-s − 0.507·35-s − 0.493·37-s − 4.74·40-s + 1.40·41-s − 1.67·43-s − 9.94·44-s − 0.884·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} + 4 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.7.b_n_e |
| 11 | $S_4\times C_2$ | \( 1 + p T + 61 T^{2} + 234 T^{3} + 61 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.11.l_cj_ja |
| 13 | $S_4\times C_2$ | \( 1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6} \) | 3.13.a_ab_abg |
| 17 | $S_4\times C_2$ | \( 1 + T + 39 T^{2} + 42 T^{3} + 39 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.17.b_bn_bq |
| 19 | $S_4\times C_2$ | \( 1 + 47 T^{2} + 4 T^{3} + 47 p T^{4} + p^{3} T^{6} \) | 3.19.a_bv_e |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 156 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ac_v_aga |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 71 T^{2} - 214 T^{3} + 71 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_ct_aig |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} + 84 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_v_dg |
| 41 | $S_4\times C_2$ | \( 1 - 9 T + 83 T^{2} - 374 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aj_df_aok |
| 43 | $S_4\times C_2$ | \( 1 + 11 T + 145 T^{2} + 866 T^{3} + 145 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.l_fp_bhi |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 111 T^{2} + 132 T^{3} + 111 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_eh_fc |
| 53 | $S_4\times C_2$ | \( 1 + 21 T + 239 T^{2} + 1910 T^{3} + 239 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.v_jf_cvm |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 211 T^{2} + 1572 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.o_id_cim |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 175 T^{2} - 850 T^{3} + 175 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ah_gt_abgs |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 171 T^{2} - 212 T^{3} + 171 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_gp_aie |
| 71 | $S_4\times C_2$ | \( 1 + 22 T + 277 T^{2} + 2484 T^{3} + 277 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.w_kr_dro |
| 73 | $S_4\times C_2$ | \( 1 + 16 T + 255 T^{2} + 2128 T^{3} + 255 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.q_jv_ddw |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 407 T^{2} + 4332 T^{3} + 407 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ba_pr_gkq |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} + 12 p T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.c_dn_bmi |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) | 3.89.s_ol_fbo |
| 97 | $S_4\times C_2$ | \( 1 + 21 T + 371 T^{2} + 3758 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.v_oh_foo |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.210328469619806186309293623023, −7.70007876145339272672595547563, −7.69579993441764150048228477360, −7.47712137994913074806558986733, −7.06003618243218109488299685193, −6.82721943116203372571817642309, −6.66622692625480793092338335561, −6.28372532299509941393636028802, −6.11581992695415511204386958984, −5.84452576302961133975493979749, −5.53938086533927244150498816698, −5.34085452700740260712226011633, −5.14353736182402394092052018844, −4.58875985804778165614455517511, −4.58545706513461456089084257968, −4.27466615243464645162382465458, −3.24704017252383920070718024178, −3.21853516905563634382777127551, −3.08201735038484098590299182150, −2.58341573525143765541493785370, −2.53151508336947485107452488907, −2.31004511769648227849935915419, −1.53138459019457292030731447648, −1.36850503802492192231121032476, −1.34827461296543348290549238663, 0, 0, 0,
1.34827461296543348290549238663, 1.36850503802492192231121032476, 1.53138459019457292030731447648, 2.31004511769648227849935915419, 2.53151508336947485107452488907, 2.58341573525143765541493785370, 3.08201735038484098590299182150, 3.21853516905563634382777127551, 3.24704017252383920070718024178, 4.27466615243464645162382465458, 4.58545706513461456089084257968, 4.58875985804778165614455517511, 5.14353736182402394092052018844, 5.34085452700740260712226011633, 5.53938086533927244150498816698, 5.84452576302961133975493979749, 6.11581992695415511204386958984, 6.28372532299509941393636028802, 6.66622692625480793092338335561, 6.82721943116203372571817642309, 7.06003618243218109488299685193, 7.47712137994913074806558986733, 7.69579993441764150048228477360, 7.70007876145339272672595547563, 8.210328469619806186309293623023
