| L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s + 2·15-s + 8·17-s + 2·19-s − 6·23-s − 25-s − 27-s − 6·29-s + 6·31-s + 33-s + 8·37-s + 6·41-s − 2·45-s + 8·47-s − 7·49-s − 8·51-s − 12·53-s + 2·55-s − 2·57-s + 4·59-s + 10·61-s − 8·67-s + 6·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 1.94·17-s + 0.458·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.07·31-s + 0.174·33-s + 1.31·37-s + 0.937·41-s − 0.298·45-s + 1.16·47-s − 49-s − 1.12·51-s − 1.64·53-s + 0.269·55-s − 0.264·57-s + 0.520·59-s + 1.28·61-s − 0.977·67-s + 0.722·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.08461748958111, −13.74181421734196, −12.81697720859127, −12.71462263146721, −12.00549386940390, −11.69645118493197, −11.32545756261222, −10.73224555863544, −10.06560454423923, −9.786308830079047, −9.307020388522559, −8.368643436900200, −7.912173849777275, −7.667234938740979, −7.186234127932165, −6.350911369875200, −5.790644184759215, −5.548762045720443, −4.725875623895526, −4.204783143168142, −3.674908098439402, −3.106904208770505, −2.369782251877704, −1.442673788168405, −0.8027998762090256, 0,
0.8027998762090256, 1.442673788168405, 2.369782251877704, 3.106904208770505, 3.674908098439402, 4.204783143168142, 4.725875623895526, 5.548762045720443, 5.790644184759215, 6.350911369875200, 7.186234127932165, 7.667234938740979, 7.912173849777275, 8.368643436900200, 9.307020388522559, 9.786308830079047, 10.06560454423923, 10.73224555863544, 11.32545756261222, 11.69645118493197, 12.00549386940390, 12.71462263146721, 12.81697720859127, 13.74181421734196, 14.08461748958111
