| L(s) = 1 | + 2·5-s − 4·7-s − 11-s + 6·13-s − 2·17-s − 4·19-s − 4·23-s − 25-s + 6·29-s − 8·35-s − 6·37-s + 6·41-s − 4·43-s + 12·47-s + 9·49-s + 2·53-s − 2·55-s + 12·59-s + 14·61-s + 12·65-s − 4·67-s + 12·71-s − 6·73-s + 4·77-s − 4·79-s + 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.301·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.35·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.79·61-s + 1.48·65-s − 0.488·67-s + 1.42·71-s − 0.702·73-s + 0.455·77-s − 0.450·79-s + 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.791442252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.791442252\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−8.288683908358154144546328543998, −7.04694782647470407191514539624, −6.51503267263782800870717672489, −5.98632558021583238878804254265, −5.49355597914156386912328033057, −4.19689350241926690371497535314, −3.66487191683449651589408155835, −2.70238009742303454701502000782, −1.95816510427660270529639249163, −0.67974140487940718393639609709,
0.67974140487940718393639609709, 1.95816510427660270529639249163, 2.70238009742303454701502000782, 3.66487191683449651589408155835, 4.19689350241926690371497535314, 5.49355597914156386912328033057, 5.98632558021583238878804254265, 6.51503267263782800870717672489, 7.04694782647470407191514539624, 8.288683908358154144546328543998
