| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 4·10-s − 3·11-s − 7·13-s − 4·16-s + 4·19-s − 4·20-s − 6·22-s − 23-s − 25-s − 14·26-s + 9·29-s − 2·31-s − 8·32-s + 8·37-s + 8·38-s − 9·41-s + 7·43-s − 6·44-s − 2·46-s − 2·50-s − 14·52-s + 6·53-s + 6·55-s + 18·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s − 0.904·11-s − 1.94·13-s − 16-s + 0.917·19-s − 0.894·20-s − 1.27·22-s − 0.208·23-s − 1/5·25-s − 2.74·26-s + 1.67·29-s − 0.359·31-s − 1.41·32-s + 1.31·37-s + 1.29·38-s − 1.40·41-s + 1.06·43-s − 0.904·44-s − 0.294·46-s − 0.282·50-s − 1.94·52-s + 0.824·53-s + 0.809·55-s + 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617772521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617772521\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.44751110549256, −12.10716094592911, −11.82974755293525, −11.34630789145923, −10.88256774901167, −10.23424416137936, −9.830635383517994, −9.453653700583465, −8.786096494285613, −8.131401591390299, −7.792107070554765, −7.412266240607182, −6.825140954168112, −6.516602941636118, −5.719416709242800, −5.260076675840297, −5.012105868859923, −4.513341537558479, −4.027199345192503, −3.556054005227398, −2.882884545385502, −2.559694938404709, −2.167354963231610, −1.037446386816605, −0.2715905937833016,
0.2715905937833016, 1.037446386816605, 2.167354963231610, 2.559694938404709, 2.882884545385502, 3.556054005227398, 4.027199345192503, 4.513341537558479, 5.012105868859923, 5.260076675840297, 5.719416709242800, 6.516602941636118, 6.825140954168112, 7.412266240607182, 7.792107070554765, 8.131401591390299, 8.786096494285613, 9.453653700583465, 9.830635383517994, 10.23424416137936, 10.88256774901167, 11.34630789145923, 11.82974755293525, 12.10716094592911, 12.44751110549256
