| L(s) = 1 | + 2·5-s − 2·17-s + 4·19-s + 23-s − 25-s + 10·29-s − 6·31-s + 2·37-s + 4·43-s − 6·47-s + 2·53-s − 2·59-s + 2·61-s − 8·71-s − 4·73-s − 4·79-s − 4·85-s + 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·115-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.485·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s + 1.85·29-s − 1.07·31-s + 0.328·37-s + 0.609·43-s − 0.875·47-s + 0.274·53-s − 0.260·59-s + 0.256·61-s − 0.949·71-s − 0.468·73-s − 0.450·79-s − 0.433·85-s + 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.186·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.56548886504642, −13.04449379475847, −12.70531091172551, −11.89712662759942, −11.80191738828001, −11.03952129066465, −10.63313421775625, −10.11442783987946, −9.682145734772813, −9.274861709197447, −8.744100679746447, −8.291362671874077, −7.613697677351211, −7.199183948581780, −6.591984461179433, −6.126459709375323, −5.679507555741655, −5.086297354742182, −4.655577647738545, −3.994548681941397, −3.319856211283103, −2.738722779208451, −2.229926950136191, −1.510429233141412, −0.9747583174937000, 0,
0.9747583174937000, 1.510429233141412, 2.229926950136191, 2.738722779208451, 3.319856211283103, 3.994548681941397, 4.655577647738545, 5.086297354742182, 5.679507555741655, 6.126459709375323, 6.591984461179433, 7.199183948581780, 7.613697677351211, 8.291362671874077, 8.744100679746447, 9.274861709197447, 9.682145734772813, 10.11442783987946, 10.63313421775625, 11.03952129066465, 11.80191738828001, 11.89712662759942, 12.70531091172551, 13.04449379475847, 13.56548886504642
