| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 4·13-s − 15-s − 6·17-s − 2·19-s − 2·21-s − 8·23-s + 25-s + 27-s + 6·29-s + 2·35-s + 6·37-s + 4·39-s − 10·41-s + 2·43-s − 45-s − 12·47-s − 3·49-s − 6·51-s + 6·53-s − 2·57-s + 8·59-s − 2·63-s − 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.640·39-s − 1.56·41-s + 0.304·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.264·57-s + 1.04·59-s − 0.251·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 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.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 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
−13.67718864497357, −13.43796967296098, −12.89997374050985, −12.58834018287716, −11.79521406937111, −11.49651897353258, −10.95995169370224, −10.32017233619688, −9.938673045500193, −9.485434607804552, −8.719829065152951, −8.411339573896540, −8.192396139506885, −7.362641666085526, −6.804938849228625, −6.287152058150946, −6.118820248600099, −5.125209298840464, −4.469701270422291, −4.061568151393645, −3.501134134867162, −3.020538435636166, −2.228098960557005, −1.780755735994646, −0.7764302250286481, 0,
0.7764302250286481, 1.780755735994646, 2.228098960557005, 3.020538435636166, 3.501134134867162, 4.061568151393645, 4.469701270422291, 5.125209298840464, 6.118820248600099, 6.287152058150946, 6.804938849228625, 7.362641666085526, 8.192396139506885, 8.411339573896540, 8.719829065152951, 9.485434607804552, 9.938673045500193, 10.32017233619688, 10.95995169370224, 11.49651897353258, 11.79521406937111, 12.58834018287716, 12.89997374050985, 13.43796967296098, 13.67718864497357
