| L(s) = 1 | − 3-s + 3·5-s + 9-s − 11-s − 13-s − 3·15-s − 7·17-s + 19-s − 7·23-s + 4·25-s − 27-s − 3·29-s + 33-s + 5·37-s + 39-s − 4·41-s − 11·43-s + 3·45-s + 7·51-s + 14·53-s − 3·55-s − 57-s + 4·59-s + 61-s − 3·65-s + 6·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.774·15-s − 1.69·17-s + 0.229·19-s − 1.45·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.174·33-s + 0.821·37-s + 0.160·39-s − 0.624·41-s − 1.67·43-s + 0.447·45-s + 0.980·51-s + 1.92·53-s − 0.404·55-s − 0.132·57-s + 0.520·59-s + 0.128·61-s − 0.372·65-s + 0.733·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.087909280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087909280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−13.42827374889065, −13.18311998123170, −12.75545638291711, −11.97909284019009, −11.61323850958400, −11.20276953942636, −10.44183608173499, −10.20988439157577, −9.734884351553537, −9.328899489905519, −8.539420702107332, −8.384171806045136, −7.395058116414764, −7.024004839952328, −6.458585106478038, −5.965169661708711, −5.593715236771728, −5.061326432555013, −4.418621227171439, −3.984694607119120, −3.072668086811406, −2.317921695400919, −2.006177284226939, −1.365740458464216, −0.3130286183966757,
0.3130286183966757, 1.365740458464216, 2.006177284226939, 2.317921695400919, 3.072668086811406, 3.984694607119120, 4.418621227171439, 5.061326432555013, 5.593715236771728, 5.965169661708711, 6.458585106478038, 7.024004839952328, 7.395058116414764, 8.384171806045136, 8.539420702107332, 9.328899489905519, 9.734884351553537, 10.20988439157577, 10.44183608173499, 11.20276953942636, 11.61323850958400, 11.97909284019009, 12.75545638291711, 13.18311998123170, 13.42827374889065
