| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 5·11-s − 2·13-s − 3·17-s − 6·19-s + 3·21-s − 8·23-s + 5·27-s + 9·29-s + 9·31-s + 5·33-s + 9·37-s + 2·39-s − 10·41-s − 6·43-s − 6·47-s + 2·49-s + 3·51-s + 2·53-s + 6·57-s + 59-s − 7·61-s + 6·63-s + 8·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.50·11-s − 0.554·13-s − 0.727·17-s − 1.37·19-s + 0.654·21-s − 1.66·23-s + 0.962·27-s + 1.67·29-s + 1.61·31-s + 0.870·33-s + 1.47·37-s + 0.320·39-s − 1.56·41-s − 0.914·43-s − 0.875·47-s + 2/7·49-s + 0.420·51-s + 0.274·53-s + 0.794·57-s + 0.130·59-s − 0.896·61-s + 0.755·63-s + 0.977·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.24155284750718, −15.02199900927693, −14.00565086130504, −13.68170067614674, −13.16594445764796, −12.50939383493803, −12.24717198262233, −11.60314878767072, −10.99086478316834, −10.39335481373335, −10.00898671108007, −9.659645721281141, −8.606568747633775, −8.229314865039628, −7.877555626326836, −6.677774589998785, −6.484366921424457, −6.091582461547951, −5.165559003766585, −4.802533853131614, −4.075801669136709, −3.130026415527061, −2.607742140493390, −2.101144610819812, −0.5752929578775308, 0,
0.5752929578775308, 2.101144610819812, 2.607742140493390, 3.130026415527061, 4.075801669136709, 4.802533853131614, 5.165559003766585, 6.091582461547951, 6.484366921424457, 6.677774589998785, 7.877555626326836, 8.229314865039628, 8.606568747633775, 9.659645721281141, 10.00898671108007, 10.39335481373335, 10.99086478316834, 11.60314878767072, 12.24717198262233, 12.50939383493803, 13.16594445764796, 13.68170067614674, 14.00565086130504, 15.02199900927693, 15.24155284750718
