| L(s) = 1 | − 2·7-s − 11-s + 2·13-s + 6·17-s − 2·19-s − 5·25-s + 6·29-s + 4·31-s + 2·37-s − 6·41-s + 10·43-s + 12·47-s − 3·49-s + 12·53-s + 12·59-s − 10·61-s − 8·67-s + 12·71-s + 14·73-s + 2·77-s − 2·79-s − 12·83-s − 4·91-s + 2·97-s + 6·101-s − 8·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s − 25-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.52·43-s + 1.75·47-s − 3/7·49-s + 1.64·53-s + 1.56·59-s − 1.28·61-s − 0.977·67-s + 1.42·71-s + 1.63·73-s + 0.227·77-s − 0.225·79-s − 1.31·83-s − 0.419·91-s + 0.203·97-s + 0.597·101-s − 0.788·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.563116816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.563116816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.527609696179096575338835429540, −8.576230685413191782049668920273, −7.88290865143984025443570728659, −6.99781897214926848513247214405, −6.09818303485559998266442512181, −5.49460624335439645618753295389, −4.26755166817314543764872138558, −3.41506653312949974796192176000, −2.43470426817624130132300754750, −0.890846737799134523556663864580,
0.890846737799134523556663864580, 2.43470426817624130132300754750, 3.41506653312949974796192176000, 4.26755166817314543764872138558, 5.49460624335439645618753295389, 6.09818303485559998266442512181, 6.99781897214926848513247214405, 7.88290865143984025443570728659, 8.576230685413191782049668920273, 9.527609696179096575338835429540
