| L(s) = 1 | − 3-s − 2·9-s − 2·11-s + 5·13-s + 4·17-s + 2·19-s − 23-s + 5·27-s − 3·29-s − 7·31-s + 2·33-s + 2·37-s − 5·39-s − 9·41-s − 4·43-s − 9·47-s − 7·49-s − 4·51-s + 6·53-s − 2·57-s + 2·61-s − 2·67-s + 69-s + 71-s − 73-s + 14·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.603·11-s + 1.38·13-s + 0.970·17-s + 0.458·19-s − 0.208·23-s + 0.962·27-s − 0.557·29-s − 1.25·31-s + 0.348·33-s + 0.328·37-s − 0.800·39-s − 1.40·41-s − 0.609·43-s − 1.31·47-s − 49-s − 0.560·51-s + 0.824·53-s − 0.264·57-s + 0.256·61-s − 0.244·67-s + 0.120·69-s + 0.118·71-s − 0.117·73-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 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.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−7.42314403120293093723401726142, −6.50857844199842191639699083912, −5.99121360965929980573080081366, −5.33701554244838769728087900507, −4.87338196650952481178472651512, −3.51743385846087025954752613129, −3.37382764401319970889447237756, −2.09785343188684447011933375493, −1.13084125980465834121328993606, 0,
1.13084125980465834121328993606, 2.09785343188684447011933375493, 3.37382764401319970889447237756, 3.51743385846087025954752613129, 4.87338196650952481178472651512, 5.33701554244838769728087900507, 5.99121360965929980573080081366, 6.50857844199842191639699083912, 7.42314403120293093723401726142
