| L(s) = 1 | + 3-s − 3·5-s − 2·9-s + 11-s + 4·13-s − 3·15-s + 6·17-s + 2·19-s − 3·23-s + 4·25-s − 5·27-s − 6·29-s + 5·31-s + 33-s + 11·37-s + 4·39-s − 6·41-s − 8·43-s + 6·45-s + 6·51-s − 6·53-s − 3·55-s + 2·57-s − 9·59-s + 10·61-s − 12·65-s − 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s + 1.45·17-s + 0.458·19-s − 0.625·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s + 0.898·31-s + 0.174·33-s + 1.80·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.894·45-s + 0.840·51-s − 0.824·53-s − 0.404·55-s + 0.264·57-s − 1.17·59-s + 1.28·61-s − 1.48·65-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.756298930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756298930\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−7.984160970521428842640406008347, −7.38510351141231969120632832405, −6.34570889225344780507838826204, −5.80750155243207811982157626971, −4.89113901035243351632315791369, −3.97411008767739240182954437363, −3.46347204407075861821751449554, −2.99165559308723446698105013962, −1.70848807533838726380425683703, −0.63715395674397493657863840222,
0.63715395674397493657863840222, 1.70848807533838726380425683703, 2.99165559308723446698105013962, 3.46347204407075861821751449554, 3.97411008767739240182954437363, 4.89113901035243351632315791369, 5.80750155243207811982157626971, 6.34570889225344780507838826204, 7.38510351141231969120632832405, 7.984160970521428842640406008347
