| L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s + 2·13-s + 2·15-s − 6·17-s + 4·23-s − 25-s + 27-s − 33-s + 10·37-s + 2·39-s − 6·41-s + 8·43-s + 2·45-s + 4·47-s − 7·49-s − 6·51-s − 6·53-s − 2·55-s − 12·59-s − 2·61-s + 4·65-s + 4·67-s + 4·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.174·33-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s + 0.583·47-s − 49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s − 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.14957254951886, −12.78146787487724, −12.66576274745607, −11.58530671283413, −11.35675595958913, −10.79436907462458, −10.46077440606690, −9.735819604385969, −9.438840088867280, −9.079717287027552, −8.495905835217241, −8.119036786005415, −7.512770560535699, −7.000779500060140, −6.433803035979424, −6.060808783154254, −5.584553183001760, −4.711370842899364, −4.612369186971774, −3.803346469388433, −3.246468800401215, −2.575942275402508, −2.237573026756895, −1.586314405379673, −0.9628398048784023, 0,
0.9628398048784023, 1.586314405379673, 2.237573026756895, 2.575942275402508, 3.246468800401215, 3.803346469388433, 4.612369186971774, 4.711370842899364, 5.584553183001760, 6.060808783154254, 6.433803035979424, 7.000779500060140, 7.512770560535699, 8.119036786005415, 8.495905835217241, 9.079717287027552, 9.438840088867280, 9.735819604385969, 10.46077440606690, 10.79436907462458, 11.35675595958913, 11.58530671283413, 12.66576274745607, 12.78146787487724, 13.14957254951886
