| L(s) = 1 | + 2·5-s + 2·7-s + 2·13-s − 4·17-s + 8·19-s + 6·23-s − 25-s + 2·29-s + 4·35-s + 2·37-s + 4·41-s + 6·47-s − 3·49-s + 6·53-s − 8·59-s + 10·61-s + 4·65-s − 8·67-s − 8·71-s − 8·73-s − 8·79-s + 12·83-s − 8·85-s − 14·89-s + 4·91-s + 16·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 1.25·23-s − 1/5·25-s + 0.371·29-s + 0.676·35-s + 0.328·37-s + 0.624·41-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s + 1.28·61-s + 0.496·65-s − 0.977·67-s − 0.949·71-s − 0.936·73-s − 0.900·79-s + 1.31·83-s − 0.867·85-s − 1.48·89-s + 0.419·91-s + 1.64·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.495967798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.495967798\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.04722897550567, −12.27372019773361, −11.78591717828099, −11.37037013829818, −11.04791809335503, −10.45496631175943, −10.08194709565278, −9.471945217352300, −9.049891772379059, −8.812351108571866, −8.097278145437874, −7.636216072752103, −7.140396536195804, −6.704350154853298, −5.992725553723790, −5.711759076126871, −5.130300604223814, −4.741569384329467, −4.128674375733905, −3.535635319160131, −2.711547293841553, −2.568268502025432, −1.512923338474491, −1.386170023739073, −0.5820669490960359,
0.5820669490960359, 1.386170023739073, 1.512923338474491, 2.568268502025432, 2.711547293841553, 3.535635319160131, 4.128674375733905, 4.741569384329467, 5.130300604223814, 5.711759076126871, 5.992725553723790, 6.704350154853298, 7.140396536195804, 7.636216072752103, 8.097278145437874, 8.812351108571866, 9.049891772379059, 9.471945217352300, 10.08194709565278, 10.45496631175943, 11.04791809335503, 11.37037013829818, 11.78591717828099, 12.27372019773361, 13.04722897550567
