| L(s) = 1 | − 3-s + 9-s + 4·11-s + 2·13-s − 2·17-s − 27-s − 6·29-s + 8·31-s − 4·33-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s + 2·51-s − 6·53-s − 8·59-s − 2·61-s − 4·67-s + 4·71-s − 10·73-s − 12·79-s + 81-s + 4·83-s + 6·87-s + 6·89-s − 8·93-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.280·51-s − 0.824·53-s − 1.04·59-s − 0.256·61-s − 0.488·67-s + 0.474·71-s − 1.17·73-s − 1.35·79-s + 1/9·81-s + 0.439·83-s + 0.643·87-s + 0.635·89-s − 0.829·93-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.787743345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787743345\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.92807737274742, −12.36664204052860, −11.92804404644721, −11.44541584136987, −11.21236910569594, −10.68289742217671, −10.10818760623217, −9.695222273543457, −9.097611671835043, −8.849578621842293, −8.225846324333534, −7.597311369468241, −7.245454570689042, −6.515960695278703, −6.211574975553266, −5.944906435397199, −5.129272725393547, −4.645019495291146, −4.155522822433315, −3.699496612554809, −3.055816142773119, −2.382640476629051, −1.566281740167079, −1.235077987490175, −0.4033479067001509,
0.4033479067001509, 1.235077987490175, 1.566281740167079, 2.382640476629051, 3.055816142773119, 3.699496612554809, 4.155522822433315, 4.645019495291146, 5.129272725393547, 5.944906435397199, 6.211574975553266, 6.515960695278703, 7.245454570689042, 7.597311369468241, 8.225846324333534, 8.849578621842293, 9.097611671835043, 9.695222273543457, 10.10818760623217, 10.68289742217671, 11.21236910569594, 11.44541584136987, 11.92804404644721, 12.36664204052860, 12.92807737274742
