| L(s) = 1 | − 5-s + 4·7-s + 13-s − 6·17-s + 4·19-s − 6·23-s + 25-s − 6·29-s + 10·31-s − 4·35-s − 10·37-s − 6·41-s + 4·43-s − 12·47-s + 9·49-s + 12·53-s + 12·59-s − 10·61-s − 65-s − 14·67-s − 16·73-s − 8·79-s − 12·83-s + 6·85-s + 6·89-s + 4·91-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s − 0.676·35-s − 1.64·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s + 1.64·53-s + 1.56·59-s − 1.28·61-s − 0.124·65-s − 1.71·67-s − 1.87·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.39786842981121962376953809417, −6.84266911055153359366788288815, −5.92027451865494408373374553114, −5.21993004899597706667595359134, −4.51303136418763468310611959071, −4.04219672261424587413536465870, −3.03782088457995639102069801656, −2.02349530187550116275460260174, −1.37587090682442614645891244145, 0,
1.37587090682442614645891244145, 2.02349530187550116275460260174, 3.03782088457995639102069801656, 4.04219672261424587413536465870, 4.51303136418763468310611959071, 5.21993004899597706667595359134, 5.92027451865494408373374553114, 6.84266911055153359366788288815, 7.39786842981121962376953809417
