| L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s − 2·11-s + 2·15-s − 3·17-s + 8·19-s + 2·21-s − 6·23-s − 4·25-s − 4·27-s + 9·29-s − 6·31-s − 4·33-s + 35-s + 3·37-s + 3·41-s − 2·43-s + 45-s − 12·47-s + 49-s − 6·51-s + 5·53-s − 2·55-s + 16·57-s + 14·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.727·17-s + 1.83·19-s + 0.436·21-s − 1.25·23-s − 4/5·25-s − 0.769·27-s + 1.67·29-s − 1.07·31-s − 0.696·33-s + 0.169·35-s + 0.493·37-s + 0.468·41-s − 0.304·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.686·53-s − 0.269·55-s + 2.11·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.28834743528042, −13.87096743936486, −13.29951788078662, −13.20991465692294, −12.30461129615425, −11.79669900609192, −11.36298540235553, −10.78641131191178, −10.01681577602248, −9.683487283732640, −9.417481505677438, −8.546323904825183, −8.177709056246895, −7.946216791582232, −7.163462606543656, −6.742049482066274, −5.881750327518207, −5.417103751037686, −4.950211665384631, −4.010616742119084, −3.713109423613344, −2.813902522163631, −2.488664534896657, −1.858397726253607, −1.116517559300810, 0,
1.116517559300810, 1.858397726253607, 2.488664534896657, 2.813902522163631, 3.713109423613344, 4.010616742119084, 4.950211665384631, 5.417103751037686, 5.881750327518207, 6.742049482066274, 7.163462606543656, 7.946216791582232, 8.177709056246895, 8.546323904825183, 9.417481505677438, 9.683487283732640, 10.01681577602248, 10.78641131191178, 11.36298540235553, 11.79669900609192, 12.30461129615425, 13.20991465692294, 13.29951788078662, 13.87096743936486, 14.28834743528042
