| L(s) = 1 | + 4·7-s − 13-s − 2·17-s + 4·19-s + 2·23-s + 6·29-s + 2·31-s + 2·37-s − 6·41-s − 12·43-s − 8·47-s + 9·49-s − 6·53-s + 4·59-s − 10·61-s + 14·67-s + 6·71-s + 6·73-s + 4·79-s − 6·83-s + 2·89-s − 4·91-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.417·23-s + 1.11·29-s + 0.359·31-s + 0.328·37-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 1.28·61-s + 1.71·67-s + 0.712·71-s + 0.702·73-s + 0.450·79-s − 0.658·83-s + 0.211·89-s − 0.419·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.62672147470741, −12.16608564104700, −11.74927849704245, −11.30062252375311, −11.13500386994259, −10.45202265295751, −10.03839726925795, −9.580470625032370, −9.071630596243206, −8.462130264389120, −8.064268679048561, −7.958406413599237, −7.176823528757061, −6.706849609181015, −6.410753488176491, −5.547933153402821, −5.104204718786259, −4.833136175474569, −4.425715077526685, −3.660952104944677, −3.159739107960043, −2.560559476990277, −1.924385180420354, −1.435989315329836, −0.9024347655522865, 0,
0.9024347655522865, 1.435989315329836, 1.924385180420354, 2.560559476990277, 3.159739107960043, 3.660952104944677, 4.425715077526685, 4.833136175474569, 5.104204718786259, 5.547933153402821, 6.410753488176491, 6.706849609181015, 7.176823528757061, 7.958406413599237, 8.064268679048561, 8.462130264389120, 9.071630596243206, 9.580470625032370, 10.03839726925795, 10.45202265295751, 11.13500386994259, 11.30062252375311, 11.74927849704245, 12.16608564104700, 12.62672147470741
