| L(s) = 1 | − 2·4-s + 3·11-s + 13-s + 4·16-s + 3·17-s + 4·19-s − 9·23-s + 6·29-s − 2·31-s + 37-s − 3·41-s − 2·43-s − 6·44-s + 6·47-s − 2·52-s + 9·53-s − 12·59-s − 5·61-s − 8·64-s + 4·67-s − 6·68-s − 9·71-s + 14·73-s − 8·76-s − 7·79-s + 15·89-s + 18·92-s + ⋯ |
| L(s) = 1 | − 4-s + 0.904·11-s + 0.277·13-s + 16-s + 0.727·17-s + 0.917·19-s − 1.87·23-s + 1.11·29-s − 0.359·31-s + 0.164·37-s − 0.468·41-s − 0.304·43-s − 0.904·44-s + 0.875·47-s − 0.277·52-s + 1.23·53-s − 1.56·59-s − 0.640·61-s − 64-s + 0.488·67-s − 0.727·68-s − 1.06·71-s + 1.63·73-s − 0.917·76-s − 0.787·79-s + 1.58·89-s + 1.87·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.79499207061884, −13.32172526708006, −12.50297529049023, −12.23728885522403, −11.86253434141123, −11.37797537621329, −10.55936941296042, −10.12173457245531, −9.877427849415252, −9.119094491682126, −8.968942177988457, −8.306919627947231, −7.721819154862641, −7.550334025484970, −6.516981622118818, −6.289608097875015, −5.519258806297435, −5.265600165344361, −4.426403530675184, −4.089248835479253, −3.525554042303703, −3.054246054099656, −2.144466373119125, −1.362751552482469, −0.8891520775563378, 0,
0.8891520775563378, 1.362751552482469, 2.144466373119125, 3.054246054099656, 3.525554042303703, 4.089248835479253, 4.426403530675184, 5.265600165344361, 5.519258806297435, 6.289608097875015, 6.516981622118818, 7.550334025484970, 7.721819154862641, 8.306919627947231, 8.968942177988457, 9.119094491682126, 9.877427849415252, 10.12173457245531, 10.55936941296042, 11.37797537621329, 11.86253434141123, 12.23728885522403, 12.50297529049023, 13.32172526708006, 13.79499207061884
