| L(s) = 1 | + 3-s − 3·7-s + 9-s − 2·13-s − 7·17-s + 19-s − 3·21-s − 4·23-s − 5·25-s + 27-s − 7·29-s + 8·31-s − 5·37-s − 2·39-s − 12·41-s − 43-s − 10·47-s + 2·49-s − 7·51-s − 4·53-s + 57-s − 9·59-s − 4·61-s − 3·63-s + 13·67-s − 4·69-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.554·13-s − 1.69·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s − 25-s + 0.192·27-s − 1.29·29-s + 1.43·31-s − 0.821·37-s − 0.320·39-s − 1.87·41-s − 0.152·43-s − 1.45·47-s + 2/7·49-s − 0.980·51-s − 0.549·53-s + 0.132·57-s − 1.17·59-s − 0.512·61-s − 0.377·63-s + 1.58·67-s − 0.481·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4458763013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4458763013\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−14.91015115865815, −14.19550710046416, −13.59791018992741, −13.40258826136125, −12.86890384399677, −12.20778728467956, −11.79393268506425, −11.13379954556316, −10.49971277730033, −9.811965060684141, −9.611807426913696, −9.094847176629819, −8.237029111943596, −8.074469972159590, −7.121221710408271, −6.693046026217668, −6.296095620968468, −5.491525399831359, −4.757244331588862, −4.166270826436127, −3.470827835096087, −3.005248526522603, −2.143420618640639, −1.705256975857461, −0.2171473618588487,
0.2171473618588487, 1.705256975857461, 2.143420618640639, 3.005248526522603, 3.470827835096087, 4.166270826436127, 4.757244331588862, 5.491525399831359, 6.296095620968468, 6.693046026217668, 7.121221710408271, 8.074469972159590, 8.237029111943596, 9.094847176629819, 9.611807426913696, 9.811965060684141, 10.49971277730033, 11.13379954556316, 11.79393268506425, 12.20778728467956, 12.86890384399677, 13.40258826136125, 13.59791018992741, 14.19550710046416, 14.91015115865815
