| L(s) = 1 | + 5-s − 2·7-s − 3·9-s − 2·11-s − 2·13-s + 2·17-s − 6·19-s − 6·23-s + 25-s + 29-s − 2·31-s − 2·35-s + 2·37-s + 2·41-s − 3·45-s − 3·49-s − 2·53-s − 2·55-s − 4·59-s + 6·61-s + 6·63-s − 2·65-s + 2·67-s + 12·71-s − 2·73-s + 4·77-s + 10·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.185·29-s − 0.359·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.447·45-s − 3/7·49-s − 0.274·53-s − 0.269·55-s − 0.520·59-s + 0.768·61-s + 0.755·63-s − 0.248·65-s + 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.455·77-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9057232864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9057232864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−7.900620103624552596900463544580, −6.88881887230169349945858403621, −6.27428584588353663082524809241, −5.75011318846858378386770813242, −5.08602239420173099719779495234, −4.18892613277567581082443284542, −3.33283671018442020666763592509, −2.57904167901603288784114442167, −1.96675500842048761121433329899, −0.42506938794757511441581961531,
0.42506938794757511441581961531, 1.96675500842048761121433329899, 2.57904167901603288784114442167, 3.33283671018442020666763592509, 4.18892613277567581082443284542, 5.08602239420173099719779495234, 5.75011318846858378386770813242, 6.27428584588353663082524809241, 6.88881887230169349945858403621, 7.900620103624552596900463544580
