| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 13-s − 14-s + 16-s − 5·17-s + 6·19-s − 20-s + 23-s − 4·25-s + 26-s − 28-s + 6·29-s + 4·31-s + 32-s − 5·34-s + 35-s + 6·37-s + 6·38-s − 40-s − 3·41-s − 10·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 1.37·19-s − 0.223·20-s + 0.208·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.857·34-s + 0.169·35-s + 0.986·37-s + 0.973·38-s − 0.158·40-s − 0.468·41-s − 1.52·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.190356728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190356728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
| 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
−14.56318521976812, −14.01351138719613, −13.48999644712579, −13.30195493068186, −12.55914502002332, −12.01774608789698, −11.61093678602861, −11.19275397135855, −10.59626401219334, −9.781812345417496, −9.671763208964625, −8.635449724563715, −8.327949145278224, −7.640357500229011, −6.936705132639505, −6.670863058294193, −5.950140173832819, −5.352161080438297, −4.759834299027249, −4.110325831209126, −3.654932151882838, −2.880567740044131, −2.419387259148884, −1.419131681991518, −0.5799060216229686,
0.5799060216229686, 1.419131681991518, 2.419387259148884, 2.880567740044131, 3.654932151882838, 4.110325831209126, 4.759834299027249, 5.352161080438297, 5.950140173832819, 6.670863058294193, 6.936705132639505, 7.640357500229011, 8.327949145278224, 8.635449724563715, 9.671763208964625, 9.781812345417496, 10.59626401219334, 11.19275397135855, 11.61093678602861, 12.01774608789698, 12.55914502002332, 13.30195493068186, 13.48999644712579, 14.01351138719613, 14.56318521976812
