| L(s) = 1 | − 2·7-s + 4·11-s + 2·13-s + 17-s − 4·19-s + 6·23-s − 5·25-s − 8·29-s + 2·31-s + 4·37-s + 2·41-s + 4·43-s + 12·47-s − 3·49-s + 6·53-s + 4·59-s + 4·61-s − 4·67-s − 6·71-s − 6·73-s − 8·77-s + 10·79-s − 12·83-s + 10·89-s − 4·91-s − 10·97-s − 2·101-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.242·17-s − 0.917·19-s + 1.25·23-s − 25-s − 1.48·29-s + 0.359·31-s + 0.657·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s + 0.512·61-s − 0.488·67-s − 0.712·71-s − 0.702·73-s − 0.911·77-s + 1.12·79-s − 1.31·83-s + 1.05·89-s − 0.419·91-s − 1.01·97-s − 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852498385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852498385\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.347569056312767746515250188347, −7.41733789783998206152741217568, −6.81371784389331450775869549206, −6.07831591814986052399743694563, −5.57381758601517529811494705055, −4.28646140842170480343947442949, −3.85415772126991626852751716580, −2.94815835250928700326092041287, −1.87782445105462940048236026457, −0.75833783388238243642544327624,
0.75833783388238243642544327624, 1.87782445105462940048236026457, 2.94815835250928700326092041287, 3.85415772126991626852751716580, 4.28646140842170480343947442949, 5.57381758601517529811494705055, 6.07831591814986052399743694563, 6.81371784389331450775869549206, 7.41733789783998206152741217568, 8.347569056312767746515250188347
