| L(s) = 1 | − 4·7-s + 2·13-s + 2·17-s + 2·19-s + 6·23-s − 5·25-s + 2·29-s + 8·31-s − 2·37-s + 6·41-s + 2·43-s + 6·47-s + 9·49-s − 4·53-s + 4·59-s + 10·61-s − 12·67-s + 10·71-s + 12·73-s − 8·79-s + 8·83-s − 2·89-s − 8·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 1.25·23-s − 25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 9/7·49-s − 0.549·53-s + 0.520·59-s + 1.28·61-s − 1.46·67-s + 1.18·71-s + 1.40·73-s − 0.900·79-s + 0.878·83-s − 0.211·89-s − 0.838·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.978532007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978532007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.13327119312354, −14.30340162897812, −13.81787019880764, −13.38268572070453, −12.89564187797195, −12.33890190525105, −11.93250875121134, −11.20685814316662, −10.70260404505796, −10.00666925206025, −9.681604666223443, −9.159041839410827, −8.549995330607279, −7.903547809742872, −7.260453712190496, −6.679522964341284, −6.213696559312712, −5.647112546510789, −5.007284069190529, −4.106366375889933, −3.613028227339055, −2.938991780370661, −2.458157909129470, −1.239878447208374, −0.5817296613379226,
0.5817296613379226, 1.239878447208374, 2.458157909129470, 2.938991780370661, 3.613028227339055, 4.106366375889933, 5.007284069190529, 5.647112546510789, 6.213696559312712, 6.679522964341284, 7.260453712190496, 7.903547809742872, 8.549995330607279, 9.159041839410827, 9.681604666223443, 10.00666925206025, 10.70260404505796, 11.20685814316662, 11.93250875121134, 12.33890190525105, 12.89564187797195, 13.38268572070453, 13.81787019880764, 14.30340162897812, 15.13327119312354
