| L(s) = 1 | + 2·3-s + 3·5-s − 7-s + 9-s + 6·11-s − 2·13-s + 6·15-s + 6·17-s + 19-s − 2·21-s − 6·23-s + 4·25-s − 4·27-s + 31-s + 12·33-s − 3·35-s + 10·37-s − 4·39-s − 9·41-s − 8·43-s + 3·45-s − 6·49-s + 12·51-s + 18·55-s + 2·57-s + 3·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 1.54·15-s + 1.45·17-s + 0.229·19-s − 0.436·21-s − 1.25·23-s + 4/5·25-s − 0.769·27-s + 0.179·31-s + 2.08·33-s − 0.507·35-s + 1.64·37-s − 0.640·39-s − 1.40·41-s − 1.21·43-s + 0.447·45-s − 6/7·49-s + 1.68·51-s + 2.42·55-s + 0.264·57-s + 0.390·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.488389140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.488389140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−9.375413238722977996205930723557, −8.485337521697436237399508177534, −7.77657858271545466639046100598, −6.69996703872711487433720579531, −6.11504682986501835847954287417, −5.24901066220413173156598692031, −3.94833404193019318212933236503, −3.23444550137290052172173020367, −2.22540825462662035823303530466, −1.37687389739098720088206839234,
1.37687389739098720088206839234, 2.22540825462662035823303530466, 3.23444550137290052172173020367, 3.94833404193019318212933236503, 5.24901066220413173156598692031, 6.11504682986501835847954287417, 6.69996703872711487433720579531, 7.77657858271545466639046100598, 8.485337521697436237399508177534, 9.375413238722977996205930723557
