| L(s) = 1 | − 2·7-s + 11-s + 6·13-s + 6·17-s + 2·19-s − 8·23-s − 5·25-s + 2·29-s − 4·31-s − 2·37-s + 10·41-s + 6·43-s + 4·47-s − 3·49-s + 4·53-s + 4·59-s + 2·61-s + 8·67-s − 12·71-s − 2·73-s − 2·77-s + 14·79-s − 4·83-s − 12·91-s + 2·97-s − 14·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.66·13-s + 1.45·17-s + 0.458·19-s − 1.66·23-s − 25-s + 0.371·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s + 0.914·43-s + 0.583·47-s − 3/7·49-s + 0.549·53-s + 0.520·59-s + 0.256·61-s + 0.977·67-s − 1.42·71-s − 0.234·73-s − 0.227·77-s + 1.57·79-s − 0.439·83-s − 1.25·91-s + 0.203·97-s − 1.39·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009080606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009080606\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.979608889626437235836563644052, −7.44052570970644272924892522269, −6.46145516855959646172528519635, −5.88973810546336648631807916054, −5.49713638385812600795324012574, −4.01846542909354983276082314716, −3.79956872154296644446779393392, −2.89345656641700241445315198519, −1.74093177182940849666471014084, −0.76253765085072849093580299450,
0.76253765085072849093580299450, 1.74093177182940849666471014084, 2.89345656641700241445315198519, 3.79956872154296644446779393392, 4.01846542909354983276082314716, 5.49713638385812600795324012574, 5.88973810546336648631807916054, 6.46145516855959646172528519635, 7.44052570970644272924892522269, 7.979608889626437235836563644052
