| L(s) = 1 | − 3-s + 5-s − 2·9-s − 3·11-s + 5·13-s − 15-s − 3·17-s + 4·19-s + 2·23-s + 25-s + 5·27-s + 2·29-s − 31-s + 3·33-s + 37-s − 5·39-s + 6·41-s + 8·43-s − 2·45-s − 4·47-s + 3·51-s + 9·53-s − 3·55-s − 4·57-s − 13·59-s + 4·61-s + 5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 0.258·15-s − 0.727·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.962·27-s + 0.371·29-s − 0.179·31-s + 0.522·33-s + 0.164·37-s − 0.800·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s − 0.583·47-s + 0.420·51-s + 1.23·53-s − 0.404·55-s − 0.529·57-s − 1.69·59-s + 0.512·61-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.261023827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261023827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.64401993156228, −12.23593629455160, −11.57037844079551, −11.23998112303947, −10.96575916554228, −10.32597080751124, −10.21121656913878, −9.282316445834272, −8.997893495023490, −8.645426832981008, −8.006827885634583, −7.564983537441427, −7.042965655737113, −6.332146704023004, −6.115948939463151, −5.537306310422191, −5.312504661105322, −4.549312596657830, −4.205529649787650, −3.280990212625933, −2.998921480729723, −2.397942481177921, −1.708434393158693, −0.9697773270753119, −0.4865967281481587,
0.4865967281481587, 0.9697773270753119, 1.708434393158693, 2.397942481177921, 2.998921480729723, 3.280990212625933, 4.205529649787650, 4.549312596657830, 5.312504661105322, 5.537306310422191, 6.115948939463151, 6.332146704023004, 7.042965655737113, 7.564983537441427, 8.006827885634583, 8.645426832981008, 8.997893495023490, 9.282316445834272, 10.21121656913878, 10.32597080751124, 10.96575916554228, 11.23998112303947, 11.57037844079551, 12.23593629455160, 12.64401993156228
