| L(s) = 1 | + 2-s − 4-s − 3·8-s + 5·11-s + 5·13-s − 16-s + 4·17-s − 2·19-s + 5·22-s − 3·23-s + 5·26-s + 10·29-s + 6·31-s + 5·32-s + 4·34-s − 5·37-s − 2·38-s + 10·41-s − 10·43-s − 5·44-s − 3·46-s − 5·47-s − 7·49-s − 5·52-s − 2·53-s + 10·58-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1.50·11-s + 1.38·13-s − 1/4·16-s + 0.970·17-s − 0.458·19-s + 1.06·22-s − 0.625·23-s + 0.980·26-s + 1.85·29-s + 1.07·31-s + 0.883·32-s + 0.685·34-s − 0.821·37-s − 0.324·38-s + 1.56·41-s − 1.52·43-s − 0.753·44-s − 0.442·46-s − 0.729·47-s − 49-s − 0.693·52-s − 0.274·53-s + 1.31·58-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991119528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991119528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−10.47476969904140521612928427933, −9.594380714160926227112344225322, −8.728922113712677747638974228180, −8.110625633965123632872532116674, −6.47087996746914035408335230406, −6.14513598061725646029734075717, −4.84886591033793439685261848050, −3.97564733739679917868066949463, −3.18705877871324552307068063652, −1.22726947298899977457345127093,
1.22726947298899977457345127093, 3.18705877871324552307068063652, 3.97564733739679917868066949463, 4.84886591033793439685261848050, 6.14513598061725646029734075717, 6.47087996746914035408335230406, 8.110625633965123632872532116674, 8.728922113712677747638974228180, 9.594380714160926227112344225322, 10.47476969904140521612928427933
