| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2·13-s + 15-s + 2·17-s + 21-s + 25-s + 27-s + 6·29-s − 4·31-s + 35-s − 2·37-s + 2·39-s + 10·41-s − 4·43-s + 45-s + 49-s + 2·51-s + 2·53-s − 4·59-s + 6·61-s + 63-s + 2·65-s + 12·67-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.520·59-s + 0.768·61-s + 0.125·63-s + 0.248·65-s + 1.46·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559128703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559128703\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.272137463304519888721719771993, −8.583915667205168907938732808044, −7.88218804424170995712718546296, −7.03203260508625422741500199905, −6.12445278441503394177450128053, −5.26872389432656365187525030939, −4.29122686155678571933750083631, −3.32010977241104683139327018187, −2.30215882443364192027548996597, −1.19019368875510050700001453681,
1.19019368875510050700001453681, 2.30215882443364192027548996597, 3.32010977241104683139327018187, 4.29122686155678571933750083631, 5.26872389432656365187525030939, 6.12445278441503394177450128053, 7.03203260508625422741500199905, 7.88218804424170995712718546296, 8.583915667205168907938732808044, 9.272137463304519888721719771993
