| L(s) = 1 | + 2·3-s − 2·5-s + 3·7-s + 9-s + 11-s − 3·13-s − 4·15-s − 17-s − 8·19-s + 6·21-s − 8·23-s − 25-s − 4·27-s + 4·29-s + 4·31-s + 2·33-s − 6·35-s + 37-s − 6·39-s + 5·41-s − 9·43-s − 2·45-s − 2·47-s + 2·49-s − 2·51-s − 12·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 1.03·15-s − 0.242·17-s − 1.83·19-s + 1.30·21-s − 1.66·23-s − 1/5·25-s − 0.769·27-s + 0.742·29-s + 0.718·31-s + 0.348·33-s − 1.01·35-s + 0.164·37-s − 0.960·39-s + 0.780·41-s − 1.37·43-s − 0.298·45-s − 0.291·47-s + 2/7·49-s − 0.280·51-s − 1.64·53-s − 0.269·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 2 | \( 1 \) | |
| 59 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.120881318670809927237785346933, −7.85863781610030413802545910886, −6.83377493183809283952372937254, −5.99414416174272671749705848848, −4.69056887329706857973197039773, −4.32285843057281811450822669469, −3.47942684195568495807908825108, −2.41048182559044059354747919286, −1.78991557117359256993946552576, 0,
1.78991557117359256993946552576, 2.41048182559044059354747919286, 3.47942684195568495807908825108, 4.32285843057281811450822669469, 4.69056887329706857973197039773, 5.99414416174272671749705848848, 6.83377493183809283952372937254, 7.85863781610030413802545910886, 8.120881318670809927237785346933
