| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 3·11-s − 2·13-s − 4·14-s + 16-s + 8·19-s + 3·22-s − 6·23-s − 2·26-s − 4·28-s + 3·29-s + 7·31-s + 32-s + 8·37-s + 8·38-s + 6·41-s + 4·43-s + 3·44-s − 6·46-s − 6·47-s + 9·49-s − 2·52-s − 9·53-s − 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.904·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.83·19-s + 0.639·22-s − 1.25·23-s − 0.392·26-s − 0.755·28-s + 0.557·29-s + 1.25·31-s + 0.176·32-s + 1.31·37-s + 1.29·38-s + 0.937·41-s + 0.609·43-s + 0.452·44-s − 0.884·46-s − 0.875·47-s + 9/7·49-s − 0.277·52-s − 1.23·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.85133120250188, −13.27106570618997, −12.75126966856155, −12.20650790756938, −11.99271513715111, −11.59256239883185, −10.83202895461802, −10.35579940929892, −9.669287536990785, −9.481683710798014, −9.192791275975474, −8.113035192533286, −7.758882267777829, −7.249875464700132, −6.561278658250390, −6.201177542224371, −5.947281686254389, −5.168895771616438, −4.458231471217059, −4.171602256661868, −3.321388023509412, −3.032072566654652, −2.555621545939142, −1.566807378466640, −0.9329921487022741, 0,
0.9329921487022741, 1.566807378466640, 2.555621545939142, 3.032072566654652, 3.321388023509412, 4.171602256661868, 4.458231471217059, 5.168895771616438, 5.947281686254389, 6.201177542224371, 6.561278658250390, 7.249875464700132, 7.758882267777829, 8.113035192533286, 9.192791275975474, 9.481683710798014, 9.669287536990785, 10.35579940929892, 10.83202895461802, 11.59256239883185, 11.99271513715111, 12.20650790756938, 12.75126966856155, 13.27106570618997, 13.85133120250188
