| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s + 3·12-s + 2·13-s + 16-s + 2·17-s − 6·18-s − 2·19-s + 23-s − 3·24-s − 2·26-s + 9·27-s − 29-s + 10·31-s − 32-s − 2·34-s + 6·36-s + 8·37-s + 2·38-s + 6·39-s − 3·41-s − 5·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s + 0.866·12-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.458·19-s + 0.208·23-s − 0.612·24-s − 0.392·26-s + 1.73·27-s − 0.185·29-s + 1.79·31-s − 0.176·32-s − 0.342·34-s + 36-s + 1.31·37-s + 0.324·38-s + 0.960·39-s − 0.468·41-s − 0.762·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.671229545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671229545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−8.743022593234522985721931678222, −8.300068839701508729561745308721, −7.78467201974518553154523249024, −6.90743881227953346358335962173, −6.14626622533803432582714969204, −4.77473381222051867477882648359, −3.77809987257108474495996208239, −3.01284848673728285364248175619, −2.20286905296666390503267896420, −1.15563015367105212658357718620,
1.15563015367105212658357718620, 2.20286905296666390503267896420, 3.01284848673728285364248175619, 3.77809987257108474495996208239, 4.77473381222051867477882648359, 6.14626622533803432582714969204, 6.90743881227953346358335962173, 7.78467201974518553154523249024, 8.300068839701508729561745308721, 8.743022593234522985721931678222
