| L(s) = 1 | − 2·7-s − 3·9-s + 4·11-s + 2·13-s + 2·17-s + 8·19-s + 2·23-s − 10·29-s − 2·31-s + 6·37-s + 10·41-s + 8·43-s − 2·47-s − 3·49-s + 10·53-s − 6·59-s − 2·61-s + 6·63-s + 2·67-s + 14·73-s − 8·77-s − 14·79-s + 9·81-s − 6·89-s − 4·91-s − 6·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 1.83·19-s + 0.417·23-s − 1.85·29-s − 0.359·31-s + 0.986·37-s + 1.56·41-s + 1.21·43-s − 0.291·47-s − 3/7·49-s + 1.37·53-s − 0.781·59-s − 0.256·61-s + 0.755·63-s + 0.244·67-s + 1.63·73-s − 0.911·77-s − 1.57·79-s + 81-s − 0.635·89-s − 0.419·91-s − 0.609·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 229 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.75866035403334, −12.26951916728079, −11.68938468824213, −11.44952939437415, −11.04788376353747, −10.60604665672452, −9.736261242097180, −9.539929732309204, −9.153685265092603, −8.875537479512369, −8.094980054968222, −7.669559744138613, −7.233985224099858, −6.764785113044694, −5.975169104333870, −5.926372417215080, −5.440825418946503, −4.806494151939097, −3.916710681005216, −3.796056539048939, −3.154793492221391, −2.768800983415399, −2.062046385227663, −1.181293100029866, −0.9029630994044640, 0,
0.9029630994044640, 1.181293100029866, 2.062046385227663, 2.768800983415399, 3.154793492221391, 3.796056539048939, 3.916710681005216, 4.806494151939097, 5.440825418946503, 5.926372417215080, 5.975169104333870, 6.764785113044694, 7.233985224099858, 7.669559744138613, 8.094980054968222, 8.875537479512369, 9.153685265092603, 9.539929732309204, 9.736261242097180, 10.60604665672452, 11.04788376353747, 11.44952939437415, 11.68938468824213, 12.26951916728079, 12.75866035403334
