| L(s) = 1 | + 3-s − 5-s + 9-s + 3·11-s + 13-s − 15-s + 7·19-s + 5·23-s + 25-s + 27-s + 6·31-s + 3·33-s − 3·37-s + 39-s + 3·41-s + 8·43-s − 45-s − 47-s − 5·53-s − 3·55-s + 7·57-s + 4·59-s − 8·61-s − 65-s + 5·69-s + 6·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 1.60·19-s + 1.04·23-s + 1/5·25-s + 0.192·27-s + 1.07·31-s + 0.522·33-s − 0.493·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s − 0.149·45-s − 0.145·47-s − 0.686·53-s − 0.404·55-s + 0.927·57-s + 0.520·59-s − 1.02·61-s − 0.124·65-s + 0.601·69-s + 0.712·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.953096724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.953096724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52191732523111, −13.99902142807266, −13.77703971879142, −13.13128295696166, −12.33640719404001, −12.21929598253949, −11.42310215173944, −11.08853890477240, −10.46604019774862, −9.658822327324212, −9.392755962913989, −8.856316741116378, −8.282481497436636, −7.636091591345390, −7.321484773002384, −6.558723473580954, −6.168448372061752, −5.185090046106149, −4.826303888064016, −3.978072200964100, −3.518509859334844, −2.969873343631468, −2.221269142351859, −1.223560828938488, −0.7946826059601921,
0.7946826059601921, 1.223560828938488, 2.221269142351859, 2.969873343631468, 3.518509859334844, 3.978072200964100, 4.826303888064016, 5.185090046106149, 6.168448372061752, 6.558723473580954, 7.321484773002384, 7.636091591345390, 8.282481497436636, 8.856316741116378, 9.392755962913989, 9.658822327324212, 10.46604019774862, 11.08853890477240, 11.42310215173944, 12.21929598253949, 12.33640719404001, 13.13128295696166, 13.77703971879142, 13.99902142807266, 14.52191732523111
