| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s + 13-s + 14-s + 16-s + 4·19-s − 22-s + 6·23-s − 5·25-s − 26-s − 28-s − 5·31-s − 32-s − 4·38-s + 7·43-s + 44-s − 6·46-s − 6·47-s − 6·49-s + 5·50-s + 52-s + 12·53-s + 56-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.917·19-s − 0.213·22-s + 1.25·23-s − 25-s − 0.196·26-s − 0.188·28-s − 0.898·31-s − 0.176·32-s − 0.648·38-s + 1.06·43-s + 0.150·44-s − 0.884·46-s − 0.875·47-s − 6/7·49-s + 0.707·50-s + 0.138·52-s + 1.64·53-s + 0.133·56-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271062 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271062 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.92856298839823, −12.61413434312371, −11.86523843053442, −11.63273020964959, −11.13668302866771, −10.70035693654356, −10.18924749391824, −9.681537137707913, −9.275544432692061, −9.000700026973588, −8.398520279939964, −7.847692010217983, −7.429707007370551, −7.006278182375118, −6.451018673576126, −6.039110058160730, −5.370398485950712, −5.075275734275852, −4.205910269073801, −3.689595844638741, −3.220231376056612, −2.635591148694651, −1.990410675799817, −1.331060191387186, −0.7942695970486551, 0,
0.7942695970486551, 1.331060191387186, 1.990410675799817, 2.635591148694651, 3.220231376056612, 3.689595844638741, 4.205910269073801, 5.075275734275852, 5.370398485950712, 6.039110058160730, 6.451018673576126, 7.006278182375118, 7.429707007370551, 7.847692010217983, 8.398520279939964, 9.000700026973588, 9.275544432692061, 9.681537137707913, 10.18924749391824, 10.70035693654356, 11.13668302866771, 11.63273020964959, 11.86523843053442, 12.61413434312371, 12.92856298839823
