| L(s) = 1 | + 3·3-s + 6·9-s + 3·11-s + 13-s − 5·17-s + 8·19-s − 2·23-s + 9·27-s − 29-s + 2·31-s + 9·33-s − 10·37-s + 3·39-s + 6·41-s + 4·43-s + 11·47-s − 15·51-s − 6·53-s + 24·57-s + 10·59-s + 10·67-s − 6·69-s − 10·73-s − 7·79-s + 9·81-s + 12·83-s − 3·87-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.904·11-s + 0.277·13-s − 1.21·17-s + 1.83·19-s − 0.417·23-s + 1.73·27-s − 0.185·29-s + 0.359·31-s + 1.56·33-s − 1.64·37-s + 0.480·39-s + 0.937·41-s + 0.609·43-s + 1.60·47-s − 2.10·51-s − 0.824·53-s + 3.17·57-s + 1.30·59-s + 1.22·67-s − 0.722·69-s − 1.17·73-s − 0.787·79-s + 81-s + 1.31·83-s − 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.297126691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.297126691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.447504145811414138752894965378, −7.52917575815651472519709129717, −7.14216994640797315588149549045, −6.27196909190688157452279416067, −5.22190557404471392361008817335, −4.17348321895597456565198399478, −3.71092698627393738833608367309, −2.87346274040664131988070140534, −2.07911953341898356527538818753, −1.13784823851991294530611763263,
1.13784823851991294530611763263, 2.07911953341898356527538818753, 2.87346274040664131988070140534, 3.71092698627393738833608367309, 4.17348321895597456565198399478, 5.22190557404471392361008817335, 6.27196909190688157452279416067, 7.14216994640797315588149549045, 7.52917575815651472519709129717, 8.447504145811414138752894965378
