| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s − 11-s − 13-s + 3·15-s − 3·17-s − 8·19-s + 3·21-s − 4·23-s + 25-s − 9·27-s + 3·29-s + 6·31-s + 3·33-s + 35-s − 8·37-s + 3·39-s + 10·41-s + 12·43-s − 6·45-s − 3·47-s + 49-s + 9·51-s + 12·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s − 0.727·17-s − 1.83·19-s + 0.654·21-s − 0.834·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s + 1.07·31-s + 0.522·33-s + 0.169·35-s − 1.31·37-s + 0.480·39-s + 1.56·41-s + 1.82·43-s − 0.894·45-s − 0.437·47-s + 1/7·49-s + 1.26·51-s + 1.64·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4810181702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4810181702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−10.29261691325996525705142259579, −9.093977484607888450831108786040, −8.082664236299139230672104932190, −7.03203429907519168701130428882, −6.39507720125995463881200025485, −5.71040747037952554894779064599, −4.62883124954248530395330550273, −4.07013749920791298026838213221, −2.31633884961707820482961774566, −0.55069141102609449826337285771,
0.55069141102609449826337285771, 2.31633884961707820482961774566, 4.07013749920791298026838213221, 4.62883124954248530395330550273, 5.71040747037952554894779064599, 6.39507720125995463881200025485, 7.03203429907519168701130428882, 8.082664236299139230672104932190, 9.093977484607888450831108786040, 10.29261691325996525705142259579
