| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s + 17-s + 4·19-s + 20-s − 22-s + 23-s + 25-s − 28-s − 2·29-s + 6·31-s + 32-s + 34-s − 35-s + 6·37-s + 4·38-s + 40-s + 6·41-s − 5·43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.208·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s + 1.07·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.762·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.20144377813054, −12.70717484536413, −12.18802349568704, −11.63255203006596, −11.39962235044066, −10.82548085100621, −10.18366137491622, −9.867064776443358, −9.558276359425474, −8.858107936510947, −8.308318394191538, −7.880442753515587, −7.259588929706593, −6.851037368840441, −6.393831807671345, −5.743567478566174, −5.491963122260177, −4.957036356628564, −4.339334402042471, −3.863779468717904, −3.217729018298315, −2.697919314797170, −2.374196298173407, −1.416863494676188, −1.011587198798824, 0,
1.011587198798824, 1.416863494676188, 2.374196298173407, 2.697919314797170, 3.217729018298315, 3.863779468717904, 4.339334402042471, 4.957036356628564, 5.491963122260177, 5.743567478566174, 6.393831807671345, 6.851037368840441, 7.259588929706593, 7.880442753515587, 8.308318394191538, 8.858107936510947, 9.558276359425474, 9.867064776443358, 10.18366137491622, 10.82548085100621, 11.39962235044066, 11.63255203006596, 12.18802349568704, 12.70717484536413, 13.20144377813054
