| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·11-s + 13-s − 14-s + 16-s − 3·17-s + 6·19-s + 2·22-s − 4·23-s + 26-s − 28-s − 2·29-s + 4·31-s + 32-s − 3·34-s − 3·37-s + 6·38-s + 5·43-s + 2·44-s − 4·46-s + 13·47-s − 6·49-s + 52-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.37·19-s + 0.426·22-s − 0.834·23-s + 0.196·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.493·37-s + 0.973·38-s + 0.762·43-s + 0.301·44-s − 0.589·46-s + 1.89·47-s − 6/7·49-s + 0.138·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.260638950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.260638950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.989285028448177576790337118054, −7.19292001055764931213529083442, −6.65809132898266319916512219383, −5.86565780952221027889121246671, −5.32574055151465704084844001474, −4.29593028424873403989464722407, −3.79071437514211007018089212321, −2.91753467694635841694144569262, −2.03126985561255364123317611578, −0.872274280960199070325328094913,
0.872274280960199070325328094913, 2.03126985561255364123317611578, 2.91753467694635841694144569262, 3.79071437514211007018089212321, 4.29593028424873403989464722407, 5.32574055151465704084844001474, 5.86565780952221027889121246671, 6.65809132898266319916512219383, 7.19292001055764931213529083442, 7.989285028448177576790337118054
