| L(s) = 1 | + 7-s + 4·11-s + 6·13-s + 17-s + 4·19-s − 4·23-s − 2·29-s − 4·31-s + 6·37-s − 6·41-s − 8·43-s + 49-s − 2·53-s − 6·61-s − 8·67-s + 16·71-s + 10·73-s + 4·77-s − 4·79-s − 4·83-s − 10·89-s + 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.274·53-s − 0.768·61-s − 0.977·67-s + 1.89·71-s + 1.17·73-s + 0.455·77-s − 0.450·79-s − 0.439·83-s − 1.05·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.32596430536268, −12.80548439056031, −12.09646219531935, −11.89276986763694, −11.28578085863096, −11.07774374327752, −10.50678054495849, −9.831826769095831, −9.492174975678200, −9.005795752893360, −8.483241993537978, −8.072867383870493, −7.626069782356439, −6.920314590112930, −6.494908101670295, −6.063896698987905, −5.517002044653871, −5.044594857490244, −4.286208285216161, −3.787757229186829, −3.522090644359540, −2.827791326986118, −1.925632862791520, −1.409045461856119, −1.051428841745390, 0,
1.051428841745390, 1.409045461856119, 1.925632862791520, 2.827791326986118, 3.522090644359540, 3.787757229186829, 4.286208285216161, 5.044594857490244, 5.517002044653871, 6.063896698987905, 6.494908101670295, 6.920314590112930, 7.626069782356439, 8.072867383870493, 8.483241993537978, 9.005795752893360, 9.492174975678200, 9.831826769095831, 10.50678054495849, 11.07774374327752, 11.28578085863096, 11.89276986763694, 12.09646219531935, 12.80548439056031, 13.32596430536268
