| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 6·11-s − 13-s − 14-s + 16-s + 7·19-s − 6·22-s − 4·23-s − 26-s − 28-s − 4·29-s − 3·31-s + 32-s − 5·37-s + 7·38-s + 2·41-s + 5·43-s − 6·44-s − 4·46-s + 4·47-s − 6·49-s − 52-s + 2·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.80·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.60·19-s − 1.27·22-s − 0.834·23-s − 0.196·26-s − 0.188·28-s − 0.742·29-s − 0.538·31-s + 0.176·32-s − 0.821·37-s + 1.13·38-s + 0.312·41-s + 0.762·43-s − 0.904·44-s − 0.589·46-s + 0.583·47-s − 6/7·49-s − 0.138·52-s + 0.274·53-s − 0.133·56-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 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.81936702474299, −13.17254108293177, −12.78249483516578, −12.45873868352219, −11.91475206958253, −11.28056539708398, −10.97624377626623, −10.34662246844320, −9.909845633903732, −9.542335835285392, −8.853541545588764, −8.161441905978156, −7.584225805664638, −7.497039828215151, −6.774591470755045, −6.127840150439247, −5.543620887785796, −5.214155106761094, −4.851604175736905, −3.885162157657854, −3.576348501642719, −2.857818196165529, −2.417109027391942, −1.810271334117579, −0.8117413209461932, 0,
0.8117413209461932, 1.810271334117579, 2.417109027391942, 2.857818196165529, 3.576348501642719, 3.885162157657854, 4.851604175736905, 5.214155106761094, 5.543620887785796, 6.127840150439247, 6.774591470755045, 7.497039828215151, 7.584225805664638, 8.161441905978156, 8.853541545588764, 9.542335835285392, 9.909845633903732, 10.34662246844320, 10.97624377626623, 11.28056539708398, 11.91475206958253, 12.45873868352219, 12.78249483516578, 13.17254108293177, 13.81936702474299
