| L(s) = 1 | + 2·7-s − 3·9-s − 2·13-s − 2·19-s − 2·23-s + 2·29-s − 8·31-s − 6·37-s + 10·41-s + 4·43-s − 4·47-s − 3·49-s + 10·53-s + 8·59-s − 6·61-s − 6·63-s − 12·67-s − 8·71-s + 14·73-s − 4·79-s + 9·81-s + 83-s + 14·89-s − 4·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 0.554·13-s − 0.458·19-s − 0.417·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 1.56·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s + 1.04·59-s − 0.768·61-s − 0.755·63-s − 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.450·79-s + 81-s + 0.109·83-s + 1.48·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321271541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321271541\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.52929736845044, −12.97405086670352, −12.40947173563694, −12.01494149685493, −11.54754181824173, −11.06020365079418, −10.63980669428323, −10.22363414557397, −9.444031693240228, −9.062818068023270, −8.604543038573588, −8.064514872702851, −7.635731945591595, −7.119970786563261, −6.513782369318780, −5.789048234615334, −5.576250285018828, −4.875040701381610, −4.432477485914040, −3.738944287260573, −3.185514502603372, −2.387262089527682, −2.080253531369984, −1.238315474219894, −0.3478747559815753,
0.3478747559815753, 1.238315474219894, 2.080253531369984, 2.387262089527682, 3.185514502603372, 3.738944287260573, 4.432477485914040, 4.875040701381610, 5.576250285018828, 5.789048234615334, 6.513782369318780, 7.119970786563261, 7.635731945591595, 8.064514872702851, 8.604543038573588, 9.062818068023270, 9.444031693240228, 10.22363414557397, 10.63980669428323, 11.06020365079418, 11.54754181824173, 12.01494149685493, 12.40947173563694, 12.97405086670352, 13.52929736845044
