| L(s) = 1 | + 2-s + 4-s + 2·5-s + 2·7-s + 8-s + 2·10-s − 2·11-s − 2·13-s + 2·14-s + 16-s + 2·20-s − 2·22-s − 23-s − 25-s − 2·26-s + 2·28-s + 2·29-s + 4·31-s + 32-s + 4·35-s + 4·37-s + 2·40-s + 6·41-s − 8·43-s − 2·44-s − 46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s + 0.353·8-s + 0.632·10-s − 0.603·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.447·20-s − 0.426·22-s − 0.208·23-s − 1/5·25-s − 0.392·26-s + 0.377·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.676·35-s + 0.657·37-s + 0.316·40-s + 0.937·41-s − 1.21·43-s − 0.301·44-s − 0.147·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.553631858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.553631858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.36328840514347, −13.02698262380314, −12.46957973279774, −11.92081750267819, −11.47244472853625, −11.12653221204369, −10.39629069989217, −10.03050532277181, −9.733539265893823, −9.012988832443363, −8.365173271358722, −7.966422679023770, −7.510392613066003, −6.805538555093374, −6.390365525027245, −5.844332813685793, −5.190750564508930, −5.066276451191606, −4.376768184882314, −3.824556397023232, −3.041916451970728, −2.458322310248853, −2.073075968185757, −1.418060747393400, −0.5930806359679145,
0.5930806359679145, 1.418060747393400, 2.073075968185757, 2.458322310248853, 3.041916451970728, 3.824556397023232, 4.376768184882314, 5.066276451191606, 5.190750564508930, 5.844332813685793, 6.390365525027245, 6.805538555093374, 7.510392613066003, 7.966422679023770, 8.365173271358722, 9.012988832443363, 9.733539265893823, 10.03050532277181, 10.39629069989217, 11.12653221204369, 11.47244472853625, 11.92081750267819, 12.46957973279774, 13.02698262380314, 13.36328840514347
