| L(s) = 1 | + 4·5-s − 2·7-s + 2·13-s − 17-s + 2·19-s − 4·23-s + 11·25-s + 6·29-s − 4·31-s − 8·35-s + 10·37-s − 6·41-s − 2·43-s − 3·49-s − 12·59-s − 2·61-s + 8·65-s + 8·67-s + 16·71-s − 6·73-s + 6·79-s − 4·83-s − 4·85-s + 16·89-s − 4·91-s + 8·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s + 0.554·13-s − 0.242·17-s + 0.458·19-s − 0.834·23-s + 11/5·25-s + 1.11·29-s − 0.718·31-s − 1.35·35-s + 1.64·37-s − 0.937·41-s − 0.304·43-s − 3/7·49-s − 1.56·59-s − 0.256·61-s + 0.992·65-s + 0.977·67-s + 1.89·71-s − 0.702·73-s + 0.675·79-s − 0.439·83-s − 0.433·85-s + 1.69·89-s − 0.419·91-s + 0.820·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−12.95634212179416, −12.62724168401747, −12.11676120127574, −11.56438115760145, −10.84006674052445, −10.71029008989589, −9.900547358774403, −9.834536800627501, −9.368125579984020, −8.955705422291595, −8.340346660331871, −7.922256841496118, −7.197713416292788, −6.559492000112381, −6.374820173164829, −5.952815609393356, −5.469602753329807, −4.903755198407096, −4.446215709289932, −3.561613698119022, −3.247577107489766, −2.473758405573544, −2.188415941807788, −1.429309649388086, −0.9684731870951970, 0,
0.9684731870951970, 1.429309649388086, 2.188415941807788, 2.473758405573544, 3.247577107489766, 3.561613698119022, 4.446215709289932, 4.903755198407096, 5.469602753329807, 5.952815609393356, 6.374820173164829, 6.559492000112381, 7.197713416292788, 7.922256841496118, 8.340346660331871, 8.955705422291595, 9.368125579984020, 9.834536800627501, 9.900547358774403, 10.71029008989589, 10.84006674052445, 11.56438115760145, 12.11676120127574, 12.62724168401747, 12.95634212179416
