| L(s) = 1 | − 2·5-s − 2·11-s + 2·13-s + 2·17-s − 4·19-s − 23-s − 25-s + 6·29-s − 2·31-s − 8·37-s − 8·41-s − 6·43-s − 2·47-s − 12·53-s + 4·55-s − 2·61-s − 4·65-s + 14·67-s − 8·71-s − 12·73-s + 4·79-s + 4·83-s − 4·85-s − 6·89-s + 8·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s − 1/5·25-s + 1.11·29-s − 0.359·31-s − 1.31·37-s − 1.24·41-s − 0.914·43-s − 0.291·47-s − 1.64·53-s + 0.539·55-s − 0.256·61-s − 0.496·65-s + 1.71·67-s − 0.949·71-s − 1.40·73-s + 0.450·79-s + 0.439·83-s − 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4490392126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4490392126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.06888224497403, −13.38491303787596, −13.03539571118087, −12.39690050878923, −11.99484537707932, −11.59184233319586, −10.93772839429099, −10.55320107754698, −10.07572205002929, −9.511363861578599, −8.706924353638349, −8.377900710183433, −7.971114776716706, −7.458072049923857, −6.658802019272226, −6.481888149559969, −5.581589478698866, −5.117477131399113, −4.495305452317997, −3.904978865582158, −3.359710448547078, −2.849577819971147, −1.929278854062712, −1.336201484506049, −0.2185568307685928,
0.2185568307685928, 1.336201484506049, 1.929278854062712, 2.849577819971147, 3.359710448547078, 3.904978865582158, 4.495305452317997, 5.117477131399113, 5.581589478698866, 6.481888149559969, 6.658802019272226, 7.458072049923857, 7.971114776716706, 8.377900710183433, 8.706924353638349, 9.511363861578599, 10.07572205002929, 10.55320107754698, 10.93772839429099, 11.59184233319586, 11.99484537707932, 12.39690050878923, 13.03539571118087, 13.38491303787596, 14.06888224497403
