| L(s) = 1 | − 3-s + 9-s + 13-s − 2·17-s + 2·19-s − 8·23-s − 5·25-s − 27-s − 6·29-s + 2·31-s + 6·37-s − 39-s + 4·43-s + 8·47-s + 2·51-s + 6·53-s − 2·57-s − 4·59-s + 2·61-s + 2·67-s + 8·69-s − 4·71-s + 2·73-s + 5·75-s − 12·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s + 0.458·19-s − 1.66·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.986·37-s − 0.160·39-s + 0.609·43-s + 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s + 0.244·67-s + 0.963·69-s − 0.474·71-s + 0.234·73-s + 0.577·75-s − 1.35·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.234080549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234080549\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.45426401285167, −13.14959696787218, −12.51101527303131, −11.97312891387234, −11.70014225562950, −11.15735281140407, −10.71948241139500, −10.10875993362286, −9.752893196069427, −9.219299577441423, −8.674173212649856, −8.021133771421935, −7.579868250145779, −7.175032376623269, −6.412004181641808, −5.901000305627581, −5.720504818399867, −4.971828107136823, −4.198015160199142, −4.028267575156641, −3.294595234057173, −2.390082231585328, −1.983185189636757, −1.158614831342795, −0.3736084768378405,
0.3736084768378405, 1.158614831342795, 1.983185189636757, 2.390082231585328, 3.294595234057173, 4.028267575156641, 4.198015160199142, 4.971828107136823, 5.720504818399867, 5.901000305627581, 6.412004181641808, 7.175032376623269, 7.579868250145779, 8.021133771421935, 8.674173212649856, 9.219299577441423, 9.752893196069427, 10.10875993362286, 10.71948241139500, 11.15735281140407, 11.70014225562950, 11.97312891387234, 12.51101527303131, 13.14959696787218, 13.45426401285167
