| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 16-s − 22-s − 5·25-s − 6·29-s − 32-s − 6·37-s − 6·41-s − 8·43-s + 44-s + 6·47-s − 7·49-s + 5·50-s − 6·53-s + 6·58-s − 8·61-s + 64-s − 6·67-s − 6·71-s + 6·73-s + 6·74-s + 10·79-s + 6·82-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s + 1/4·16-s − 0.213·22-s − 25-s − 1.11·29-s − 0.176·32-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.150·44-s + 0.875·47-s − 49-s + 0.707·50-s − 0.824·53-s + 0.787·58-s − 1.02·61-s + 1/8·64-s − 0.733·67-s − 0.712·71-s + 0.702·73-s + 0.697·74-s + 1.12·79-s + 0.662·82-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8344656572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8344656572\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.19312134803893, −14.56982114145891, −13.91356642120753, −13.50722848033325, −12.82139083887993, −12.27743566056343, −11.66325425958337, −11.36612686601078, −10.62798437765331, −10.18098321637367, −9.604294510919338, −9.101275059062642, −8.598841905713389, −7.904644562815235, −7.528654425472685, −6.796362367773326, −6.338165465095913, −5.635064018693378, −5.055811327660115, −4.229793774056984, −3.519692011369084, −2.969148254978724, −1.886757638127330, −1.598900863277372, −0.3750181720386548,
0.3750181720386548, 1.598900863277372, 1.886757638127330, 2.969148254978724, 3.519692011369084, 4.229793774056984, 5.055811327660115, 5.635064018693378, 6.338165465095913, 6.796362367773326, 7.528654425472685, 7.904644562815235, 8.598841905713389, 9.101275059062642, 9.604294510919338, 10.18098321637367, 10.62798437765331, 11.36612686601078, 11.66325425958337, 12.27743566056343, 12.82139083887993, 13.50722848033325, 13.91356642120753, 14.56982114145891, 15.19312134803893
