| L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 4·13-s − 2·17-s − 4·19-s − 21-s − 7·23-s + 27-s − 9·29-s − 2·31-s + 33-s − 37-s − 4·39-s + 8·41-s + 9·43-s − 4·47-s + 49-s − 2·51-s − 6·53-s − 4·57-s + 4·59-s + 4·61-s − 63-s − 9·67-s − 7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.45·23-s + 0.192·27-s − 1.67·29-s − 0.359·31-s + 0.174·33-s − 0.164·37-s − 0.640·39-s + 1.24·41-s + 1.37·43-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.512·61-s − 0.125·63-s − 1.09·67-s − 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.849859088685116549969651186548, −7.85420867657145258328821942784, −7.31982183895857020571579368212, −6.39586265526965700202153455080, −5.60749696226775632351864214093, −4.42103072997298117901358363763, −3.83149616775640479569023732959, −2.63853935755344161562829558920, −1.86849743506491181695892823432, 0,
1.86849743506491181695892823432, 2.63853935755344161562829558920, 3.83149616775640479569023732959, 4.42103072997298117901358363763, 5.60749696226775632351864214093, 6.39586265526965700202153455080, 7.31982183895857020571579368212, 7.85420867657145258328821942784, 8.849859088685116549969651186548
