| L(s) = 1 | − 3-s − 4·5-s − 2·7-s + 9-s − 4·13-s + 4·15-s − 2·17-s + 4·19-s + 2·21-s − 6·23-s + 11·25-s − 27-s + 6·29-s − 4·31-s + 8·35-s − 2·37-s + 4·39-s − 6·41-s + 8·43-s − 4·45-s − 10·47-s − 3·49-s + 2·51-s − 12·53-s − 4·57-s − 8·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s − 0.596·45-s − 1.45·47-s − 3/7·49-s + 0.280·51-s − 1.64·53-s − 0.529·57-s − 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−16.75199081851267, −16.28272737799618, −15.88056385479245, −15.55064256768918, −14.82560142242255, −14.27488420382222, −13.52579861028912, −12.68370158403257, −12.26953340210574, −11.97671463764802, −11.27766505245713, −10.85570296891418, −9.920266811190632, −9.654631866288956, −8.639510000465844, −8.086118540590009, −7.392585089103402, −7.030539555470152, −6.314197632961270, −5.477367978109670, −4.595246988928372, −4.290969934888775, −3.326454111153154, −2.858659362367730, −1.469594719271071, 0, 0,
1.469594719271071, 2.858659362367730, 3.326454111153154, 4.290969934888775, 4.595246988928372, 5.477367978109670, 6.314197632961270, 7.030539555470152, 7.392585089103402, 8.086118540590009, 8.639510000465844, 9.654631866288956, 9.920266811190632, 10.85570296891418, 11.27766505245713, 11.97671463764802, 12.26953340210574, 12.68370158403257, 13.52579861028912, 14.27488420382222, 14.82560142242255, 15.55064256768918, 15.88056385479245, 16.28272737799618, 16.75199081851267
