| L(s) = 1 | − 2·7-s + 2·11-s + 6·13-s − 2·17-s − 4·23-s + 8·31-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s − 3·49-s − 6·53-s + 10·59-s + 2·61-s + 8·67-s + 12·71-s − 4·73-s − 4·77-s − 4·83-s + 10·89-s − 12·91-s − 8·97-s + 8·101-s + 14·103-s + 12·107-s + 10·109-s − 6·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s + 1.66·13-s − 0.485·17-s − 0.834·23-s + 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 1.30·59-s + 0.256·61-s + 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.455·77-s − 0.439·83-s + 1.05·89-s − 1.25·91-s − 0.812·97-s + 0.796·101-s + 1.37·103-s + 1.16·107-s + 0.957·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827283094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827283094\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 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 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.475936253839687637197354589621, −8.027434385497821411634796503445, −6.79000112596355133531174216421, −6.40073667901101696810712029668, −5.77273226580484918255644742694, −4.61467729663053189068781244044, −3.81836356795852594413002322118, −3.16702274094000926250632813614, −1.96622520036040824428794557555, −0.807947313312952264842686478028,
0.807947313312952264842686478028, 1.96622520036040824428794557555, 3.16702274094000926250632813614, 3.81836356795852594413002322118, 4.61467729663053189068781244044, 5.77273226580484918255644742694, 6.40073667901101696810712029668, 6.79000112596355133531174216421, 8.027434385497821411634796503445, 8.475936253839687637197354589621
