| L(s) = 1 | − 2·7-s + 3·11-s − 13-s − 6·17-s − 2·19-s − 3·23-s − 6·29-s + 4·31-s − 7·37-s − 2·43-s + 3·47-s − 3·49-s + 6·53-s + 15·59-s + 5·61-s − 2·67-s + 9·71-s + 2·73-s − 6·77-s + 10·79-s − 12·83-s − 18·89-s + 2·91-s + 17·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.625·23-s − 1.11·29-s + 0.718·31-s − 1.15·37-s − 0.304·43-s + 0.437·47-s − 3/7·49-s + 0.824·53-s + 1.95·59-s + 0.640·61-s − 0.244·67-s + 1.06·71-s + 0.234·73-s − 0.683·77-s + 1.12·79-s − 1.31·83-s − 1.90·89-s + 0.209·91-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286229907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286229907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−16.53568106508067, −15.96868761405305, −15.33465294245053, −14.98036723053488, −14.10165630923641, −13.77889692455548, −12.88935668618604, −12.74371850555879, −11.77358233927982, −11.46400489474048, −10.69859991297342, −9.983228584976558, −9.570773298395624, −8.737824181667573, −8.508166951823503, −7.410648338717284, −6.820534096312699, −6.395979278972420, −5.663873197193816, −4.801219289791506, −4.017590896284804, −3.548824924369431, −2.464509753915348, −1.830097451895276, −0.5088746904339740,
0.5088746904339740, 1.830097451895276, 2.464509753915348, 3.548824924369431, 4.017590896284804, 4.801219289791506, 5.663873197193816, 6.395979278972420, 6.820534096312699, 7.410648338717284, 8.508166951823503, 8.737824181667573, 9.570773298395624, 9.983228584976558, 10.69859991297342, 11.46400489474048, 11.77358233927982, 12.74371850555879, 12.88935668618604, 13.77889692455548, 14.10165630923641, 14.98036723053488, 15.33465294245053, 15.96868761405305, 16.53568106508067
