| 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
| \(L(1)\) |
\(\approx\) |
\(2.661203843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661203843\) |
| \(L(\frac{3}{2})\) |
|
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.ac |
| 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.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 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.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 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
−8.098635818276792269264128027492, −7.46807807596193722223289663417, −6.55969792597164920396527926365, −5.87412996828179799358291335242, −5.22586757235778997067135765525, −4.43377069596031381327397780689, −3.58239053787350433227994140979, −2.87043661698058989228588247649, −1.64743961772857105245471961438, −0.931831124352631770674309179721,
0.931831124352631770674309179721, 1.64743961772857105245471961438, 2.87043661698058989228588247649, 3.58239053787350433227994140979, 4.43377069596031381327397780689, 5.22586757235778997067135765525, 5.87412996828179799358291335242, 6.55969792597164920396527926365, 7.46807807596193722223289663417, 8.098635818276792269264128027492
