| L(s) = 1 | + 7-s − 3·9-s + 3·11-s − 3·13-s − 8·17-s + 7·19-s + 23-s + 7·29-s − 10·31-s + 4·37-s + 11·41-s − 5·43-s + 10·47-s − 6·49-s − 6·53-s + 8·59-s − 8·61-s − 3·63-s − 10·71-s − 11·73-s + 3·77-s − 3·79-s + 9·81-s − 83-s − 6·89-s − 3·91-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.904·11-s − 0.832·13-s − 1.94·17-s + 1.60·19-s + 0.208·23-s + 1.29·29-s − 1.79·31-s + 0.657·37-s + 1.71·41-s − 0.762·43-s + 1.45·47-s − 6/7·49-s − 0.824·53-s + 1.04·59-s − 1.02·61-s − 0.377·63-s − 1.18·71-s − 1.28·73-s + 0.341·77-s − 0.337·79-s + 81-s − 0.109·83-s − 0.635·89-s − 0.314·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.702961856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702961856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.52221328615588233917508597142, −7.20592228589240338008164473453, −6.27424556760251389245309732166, −5.74694853373776971122731199887, −4.83652356343344130769318993274, −4.39295040746789468527646030817, −3.35120543394723501386938522076, −2.63742678188276193761409930978, −1.79983053527845682377472927869, −0.61859271101847184395776720925,
0.61859271101847184395776720925, 1.79983053527845682377472927869, 2.63742678188276193761409930978, 3.35120543394723501386938522076, 4.39295040746789468527646030817, 4.83652356343344130769318993274, 5.74694853373776971122731199887, 6.27424556760251389245309732166, 7.20592228589240338008164473453, 7.52221328615588233917508597142
