| L(s) = 1 | − 3·5-s + 5·7-s − 2·11-s + 4·13-s + 4·17-s − 5·19-s + 4·23-s + 4·25-s + 10·29-s + 31-s − 15·35-s + 6·37-s + 5·41-s + 2·43-s − 4·47-s + 18·49-s − 12·53-s + 6·55-s − 5·59-s + 8·61-s − 12·65-s + 12·67-s + 9·71-s − 10·73-s − 10·77-s + 2·79-s − 10·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.88·7-s − 0.603·11-s + 1.10·13-s + 0.970·17-s − 1.14·19-s + 0.834·23-s + 4/5·25-s + 1.85·29-s + 0.179·31-s − 2.53·35-s + 0.986·37-s + 0.780·41-s + 0.304·43-s − 0.583·47-s + 18/7·49-s − 1.64·53-s + 0.809·55-s − 0.650·59-s + 1.02·61-s − 1.48·65-s + 1.46·67-s + 1.06·71-s − 1.17·73-s − 1.13·77-s + 0.225·79-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.477641087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.477641087\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−15.77395460795567, −15.28344146876865, −14.72246978070341, −14.30984513510127, −13.78844594469297, −12.87996796912169, −12.49800445460505, −11.76703507092597, −11.34238895614193, −10.90134827874637, −10.55882263209310, −9.628479309018994, −8.633571097224921, −8.273942444379003, −8.019281792176356, −7.449641063473772, −6.650642578546117, −5.873215379274898, −5.065387900020348, −4.547058813104708, −4.092691719014116, −3.243004896111122, −2.440302601860736, −1.379446564343012, −0.7460351217404768,
0.7460351217404768, 1.379446564343012, 2.440302601860736, 3.243004896111122, 4.092691719014116, 4.547058813104708, 5.065387900020348, 5.873215379274898, 6.650642578546117, 7.449641063473772, 8.019281792176356, 8.273942444379003, 8.633571097224921, 9.628479309018994, 10.55882263209310, 10.90134827874637, 11.34238895614193, 11.76703507092597, 12.49800445460505, 12.87996796912169, 13.78844594469297, 14.30984513510127, 14.72246978070341, 15.28344146876865, 15.77395460795567
