| L(s) = 1 | − 3-s + 9-s − 11-s − 2·13-s − 2·17-s − 4·19-s − 27-s + 2·29-s + 33-s − 2·37-s + 2·39-s + 2·41-s + 12·43-s + 8·47-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s − 12·59-s − 6·61-s − 4·67-s + 6·73-s + 16·79-s + 81-s − 4·83-s − 2·87-s + 10·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.192·27-s + 0.371·29-s + 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.768·61-s − 0.488·67-s + 0.702·73-s + 1.80·79-s + 1/9·81-s − 0.439·83-s − 0.214·87-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031450436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031450436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.47009010195324, −13.89431325866327, −13.45206617798008, −12.78835475813091, −12.37876319181358, −12.05146963341283, −11.31469608817109, −10.70334858609611, −10.61120381309907, −9.877073160751952, −9.155686307585815, −8.946741565435996, −7.992588706748152, −7.689154755757491, −6.987786457942602, −6.452624594723850, −5.955059823521374, −5.340499193700934, −4.685513230069237, −4.284744273204795, −3.565838672565764, −2.639677933860693, −2.206999979572408, −1.279182617856330, −0.3801190453498442,
0.3801190453498442, 1.279182617856330, 2.206999979572408, 2.639677933860693, 3.565838672565764, 4.284744273204795, 4.685513230069237, 5.340499193700934, 5.955059823521374, 6.452624594723850, 6.987786457942602, 7.689154755757491, 7.992588706748152, 8.946741565435996, 9.155686307585815, 9.877073160751952, 10.61120381309907, 10.70334858609611, 11.31469608817109, 12.05146963341283, 12.37876319181358, 12.78835475813091, 13.45206617798008, 13.89431325866327, 14.47009010195324
