| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s − 11-s + 5·13-s − 14-s + 16-s + 6·17-s + 3·18-s + 20-s + 22-s − 4·23-s + 25-s − 5·26-s + 28-s + 4·29-s + 5·31-s − 32-s − 6·34-s + 35-s − 3·36-s + 4·37-s − 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.980·26-s + 0.188·28-s + 0.742·29-s + 0.898·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s − 1/2·36-s + 0.657·37-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.677441745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677441745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.21739560375231, −12.56049825318605, −12.05364681213320, −11.63493393373260, −11.22575499234613, −10.77499592325172, −10.14709561770212, −9.954198245556523, −9.369870083829164, −8.660697843856550, −8.401851040798892, −8.048835306090936, −7.580844702926669, −6.808821657882470, −6.216364269783600, −6.021780810978718, −5.330942971952645, −5.035873586658402, −4.031267705190453, −3.581613221132330, −2.924746784431563, −2.439832533586472, −1.757304482815125, −1.033406388964773, −0.6167540013073078,
0.6167540013073078, 1.033406388964773, 1.757304482815125, 2.439832533586472, 2.924746784431563, 3.581613221132330, 4.031267705190453, 5.035873586658402, 5.330942971952645, 6.021780810978718, 6.216364269783600, 6.808821657882470, 7.580844702926669, 8.048835306090936, 8.401851040798892, 8.660697843856550, 9.369870083829164, 9.954198245556523, 10.14709561770212, 10.77499592325172, 11.22575499234613, 11.63493393373260, 12.05364681213320, 12.56049825318605, 13.21739560375231
