| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 12-s + 4·13-s − 2·14-s + 16-s + 6·17-s − 18-s − 2·21-s + 24-s − 4·26-s − 27-s + 2·28-s − 8·29-s − 32-s − 6·34-s + 36-s + 4·37-s − 4·39-s + 10·41-s + 2·42-s + 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.436·21-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.657·37-s − 0.640·39-s + 1.56·41-s + 0.308·42-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008513367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008513367\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.11910469986257, −12.84635418024250, −12.45241194311949, −11.63077572580372, −11.42015771714091, −11.02065934465222, −10.63741346539867, −9.939229966097099, −9.602570990026634, −9.082138680949599, −8.455405207303205, −8.001071952020820, −7.595462924110245, −7.154942361853709, −6.453149888847732, −5.847664453421243, −5.610444121636270, −5.051676923732665, −4.130470763069572, −3.891727909906307, −3.073167357044026, −2.394476959141629, −1.594836458090972, −1.167412696747786, −0.5543942938138153,
0.5543942938138153, 1.167412696747786, 1.594836458090972, 2.394476959141629, 3.073167357044026, 3.891727909906307, 4.130470763069572, 5.051676923732665, 5.610444121636270, 5.847664453421243, 6.453149888847732, 7.154942361853709, 7.595462924110245, 8.001071952020820, 8.455405207303205, 9.082138680949599, 9.602570990026634, 9.939229966097099, 10.63741346539867, 11.02065934465222, 11.42015771714091, 11.63077572580372, 12.45241194311949, 12.84635418024250, 13.11910469986257
