| L(s) = 1 | + 7-s + 4·11-s + 2·13-s − 17-s + 4·19-s − 6·29-s + 8·31-s + 6·37-s − 2·41-s + 8·43-s − 8·47-s + 49-s − 6·53-s − 10·61-s − 16·67-s + 8·71-s + 10·73-s + 4·77-s + 4·79-s − 12·83-s + 2·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.28·61-s − 1.95·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s + 0.211·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.31121808159256, −12.70871173097956, −12.27242221123338, −11.66936491618815, −11.51199206347532, −10.93369453776080, −10.56987099703752, −9.785821906853007, −9.408589735984217, −9.166703709496255, −8.466968660405474, −8.005640478823655, −7.624316994955389, −6.944113986791636, −6.544304001911459, −5.980689441707036, −5.633855309810715, −4.830624281609388, −4.442658252919557, −3.918816773751047, −3.332806647410345, −2.811955219249707, −2.050921739002346, −1.328795080843550, −1.050545425280313, 0,
1.050545425280313, 1.328795080843550, 2.050921739002346, 2.811955219249707, 3.332806647410345, 3.918816773751047, 4.442658252919557, 4.830624281609388, 5.633855309810715, 5.980689441707036, 6.544304001911459, 6.944113986791636, 7.624316994955389, 8.005640478823655, 8.466968660405474, 9.166703709496255, 9.408589735984217, 9.785821906853007, 10.56987099703752, 10.93369453776080, 11.51199206347532, 11.66936491618815, 12.27242221123338, 12.70871173097956, 13.31121808159256
