| L(s) = 1 | − 2·5-s − 2·11-s + 13-s + 2·17-s + 5·19-s − 2·23-s − 25-s + 5·31-s − 6·37-s − 12·41-s − 5·43-s − 12·47-s − 7·49-s + 10·53-s + 4·55-s − 14·59-s + 7·61-s − 2·65-s − 67-s + 2·71-s − 11·73-s + 79-s + 2·83-s − 4·85-s − 12·89-s − 10·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.14·19-s − 0.417·23-s − 1/5·25-s + 0.898·31-s − 0.986·37-s − 1.87·41-s − 0.762·43-s − 1.75·47-s − 49-s + 1.37·53-s + 0.539·55-s − 1.82·59-s + 0.896·61-s − 0.248·65-s − 0.122·67-s + 0.237·71-s − 1.28·73-s + 0.112·79-s + 0.219·83-s − 0.433·85-s − 1.27·89-s − 1.02·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−8.535302844262637931596095163808, −8.078892481701858324741008236878, −7.34402686778627599210605180375, −6.52236343024549869265581498241, −5.45824400962735163329901483843, −4.76087310144670533785104116013, −3.65621076335288866249845789169, −3.02907560406829433398625957769, −1.54470329321546224005103129311, 0,
1.54470329321546224005103129311, 3.02907560406829433398625957769, 3.65621076335288866249845789169, 4.76087310144670533785104116013, 5.45824400962735163329901483843, 6.52236343024549869265581498241, 7.34402686778627599210605180375, 8.078892481701858324741008236878, 8.535302844262637931596095163808
