| L(s) = 1 | − 3-s + 3·5-s + 5·7-s − 2·9-s + 4·13-s − 3·15-s − 6·17-s − 19-s − 5·21-s − 2·23-s + 4·25-s + 5·27-s + 5·29-s + 10·31-s + 15·35-s + 2·37-s − 4·39-s − 9·41-s + 6·43-s − 6·45-s − 4·47-s + 18·49-s + 6·51-s + 3·53-s + 57-s − 59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.88·7-s − 2/3·9-s + 1.10·13-s − 0.774·15-s − 1.45·17-s − 0.229·19-s − 1.09·21-s − 0.417·23-s + 4/5·25-s + 0.962·27-s + 0.928·29-s + 1.79·31-s + 2.53·35-s + 0.328·37-s − 0.640·39-s − 1.40·41-s + 0.914·43-s − 0.894·45-s − 0.583·47-s + 18/7·49-s + 0.840·51-s + 0.412·53-s + 0.132·57-s − 0.130·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536628236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536628236\) |
| \(L(\frac{3}{2})\) |
|
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 | 2 | \( 1 \) | |
| 59 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
| 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
−8.450801721382217519805244337818, −8.079694700721714532656923212077, −6.67739937821324708030743647069, −6.27333337035623959503787744965, −5.48089430918068205456071344524, −4.89528077648694533881216988694, −4.18766984739172873851697611340, −2.64841313526652542197079753824, −1.92312501943787052919127811999, −1.02088343698404240159174520960,
1.02088343698404240159174520960, 1.92312501943787052919127811999, 2.64841313526652542197079753824, 4.18766984739172873851697611340, 4.89528077648694533881216988694, 5.48089430918068205456071344524, 6.27333337035623959503787744965, 6.67739937821324708030743647069, 8.079694700721714532656923212077, 8.450801721382217519805244337818
