| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 2·11-s − 6·13-s − 14-s − 16-s + 8·17-s + 19-s + 2·22-s − 6·23-s − 5·25-s + 6·26-s − 28-s + 6·29-s − 5·32-s − 8·34-s − 2·37-s − 38-s + 2·41-s − 4·43-s + 2·44-s + 6·46-s − 2·47-s + 49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.603·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 1.94·17-s + 0.229·19-s + 0.426·22-s − 1.25·23-s − 25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.883·32-s − 1.37·34-s − 0.328·37-s − 0.162·38-s + 0.312·41-s − 0.609·43-s + 0.301·44-s + 0.884·46-s − 0.291·47-s + 1/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−9.531956485631663698628909214600, −8.430578954476640378685449446378, −7.73615940684089542205213733052, −7.37777819527216520176153550324, −5.81925724207397802821543968215, −5.06660425760071494184917234451, −4.22513932740494609556728886018, −2.89641997824688480943384536269, −1.54904966216975383112386801468, 0,
1.54904966216975383112386801468, 2.89641997824688480943384536269, 4.22513932740494609556728886018, 5.06660425760071494184917234451, 5.81925724207397802821543968215, 7.37777819527216520176153550324, 7.73615940684089542205213733052, 8.430578954476640378685449446378, 9.531956485631663698628909214600
