| L(s) = 1 | + 3·5-s − 7-s − 4·11-s − 3·13-s + 4·17-s − 19-s − 4·23-s + 4·25-s + 6·31-s − 3·35-s + 37-s − 5·41-s + 2·47-s − 6·49-s + 2·53-s − 12·55-s − 10·59-s + 2·61-s − 9·65-s + 12·67-s − 5·71-s + 7·73-s + 4·77-s − 10·79-s + 7·83-s + 12·85-s + 13·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 1.20·11-s − 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s + 1.07·31-s − 0.507·35-s + 0.164·37-s − 0.780·41-s + 0.291·47-s − 6/7·49-s + 0.274·53-s − 1.61·55-s − 1.30·59-s + 0.256·61-s − 1.11·65-s + 1.46·67-s − 0.593·71-s + 0.819·73-s + 0.455·77-s − 1.12·79-s + 0.768·83-s + 1.30·85-s + 1.37·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−7.59696751043267290119143799624, −6.57135455453658551617167329148, −6.13648532677140579958744272778, −5.28679021832354299011333699938, −5.00085972716244683353419500975, −3.84297471398396674405942497334, −2.79857744411560880003140089736, −2.37444494347776736142497194654, −1.38055365845120531923139519541, 0,
1.38055365845120531923139519541, 2.37444494347776736142497194654, 2.79857744411560880003140089736, 3.84297471398396674405942497334, 5.00085972716244683353419500975, 5.28679021832354299011333699938, 6.13648532677140579958744272778, 6.57135455453658551617167329148, 7.59696751043267290119143799624
