| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 2·11-s − 2·13-s − 2·15-s + 6·17-s + 6·19-s − 8·21-s − 4·23-s + 25-s − 4·27-s + 29-s − 2·31-s − 4·33-s + 4·35-s + 2·37-s − 4·39-s − 6·41-s − 6·43-s − 45-s + 6·47-s + 9·49-s + 12·51-s − 2·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 1.37·19-s − 1.74·21-s − 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.185·29-s − 0.359·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s + 1.68·51-s − 0.274·53-s + 0.269·55-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
| \(L(1)\) |
\(\approx\) |
\(1.770926085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770926085\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−7.70326478963226865999829100657, −7.33607261103935391982988400204, −6.41648637742542767727409476702, −5.67009841274337685604884208838, −4.96810759046459358627910984310, −3.81772207985290911566811238290, −3.24005724580949367693219876865, −2.98380024159108736916036934945, −1.96599562798330226815226019498, −0.57425133540426733942746125977,
0.57425133540426733942746125977, 1.96599562798330226815226019498, 2.98380024159108736916036934945, 3.24005724580949367693219876865, 3.81772207985290911566811238290, 4.96810759046459358627910984310, 5.67009841274337685604884208838, 6.41648637742542767727409476702, 7.33607261103935391982988400204, 7.70326478963226865999829100657
