| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s + 2·13-s + 17-s + 21-s + 4·23-s − 27-s − 6·29-s − 2·31-s + 2·33-s + 6·37-s − 2·39-s − 2·41-s + 2·43-s − 2·47-s + 49-s − 51-s − 10·53-s − 10·61-s − 63-s + 10·67-s − 4·69-s + 2·71-s + 14·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.242·17-s + 0.218·21-s + 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 0.986·37-s − 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.291·47-s + 1/7·49-s − 0.140·51-s − 1.37·53-s − 1.28·61-s − 0.125·63-s + 1.22·67-s − 0.481·69-s + 0.237·71-s + 1.63·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.19065399138413, −14.80700144492361, −14.08010894424787, −13.48456199654634, −13.03005947160967, −12.62845555305970, −12.10059785506294, −11.33552133413772, −10.97960505230449, −10.60645820944841, −9.811181160159895, −9.398218955225325, −8.885155154916274, −7.928432906407905, −7.770567683112612, −6.880019451500589, −6.456176173616909, −5.803206506523039, −5.278895515693994, −4.727702489425582, −3.910061246565332, −3.349675156070289, −2.608714928587392, −1.747101380075048, −0.9085194513925978, 0,
0.9085194513925978, 1.747101380075048, 2.608714928587392, 3.349675156070289, 3.910061246565332, 4.727702489425582, 5.278895515693994, 5.803206506523039, 6.456176173616909, 6.880019451500589, 7.770567683112612, 7.928432906407905, 8.885155154916274, 9.398218955225325, 9.811181160159895, 10.60645820944841, 10.97960505230449, 11.33552133413772, 12.10059785506294, 12.62845555305970, 13.03005947160967, 13.48456199654634, 14.08010894424787, 14.80700144492361, 15.19065399138413
