| L(s) = 1 | + 2·3-s + 9-s + 2·13-s + 3·17-s + 8·19-s − 9·23-s − 4·27-s + 6·29-s − 5·31-s + 8·37-s + 4·39-s − 3·41-s + 10·43-s − 3·47-s + 6·51-s + 6·53-s + 16·57-s + 12·59-s + 4·61-s − 2·67-s − 18·69-s + 9·71-s + 10·73-s − 5·79-s − 11·81-s + 6·83-s + 12·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.554·13-s + 0.727·17-s + 1.83·19-s − 1.87·23-s − 0.769·27-s + 1.11·29-s − 0.898·31-s + 1.31·37-s + 0.640·39-s − 0.468·41-s + 1.52·43-s − 0.437·47-s + 0.840·51-s + 0.824·53-s + 2.11·57-s + 1.56·59-s + 0.512·61-s − 0.244·67-s − 2.16·69-s + 1.06·71-s + 1.17·73-s − 0.562·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.602484806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.602484806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.08782023688230, −13.66394621764116, −13.20664523943302, −12.57514110868121, −11.98704227604594, −11.63293830092663, −11.08123532437031, −10.31482407691530, −9.878987456368986, −9.462939434123142, −9.014618536707159, −8.261458703251471, −8.034148664324415, −7.570014450444794, −6.983584640981220, −6.222095763732707, −5.657318317795060, −5.266595664534405, −4.337359540653412, −3.709500437286925, −3.455974871884213, −2.634314563099769, −2.235929087698000, −1.343037567258514, −0.6819396798182991,
0.6819396798182991, 1.343037567258514, 2.235929087698000, 2.634314563099769, 3.455974871884213, 3.709500437286925, 4.337359540653412, 5.266595664534405, 5.657318317795060, 6.222095763732707, 6.983584640981220, 7.570014450444794, 8.034148664324415, 8.261458703251471, 9.014618536707159, 9.462939434123142, 9.878987456368986, 10.31482407691530, 11.08123532437031, 11.63293830092663, 11.98704227604594, 12.57514110868121, 13.20664523943302, 13.66394621764116, 14.08782023688230
