| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 5·13-s + 14-s + 16-s − 4·19-s − 20-s − 22-s + 6·23-s + 25-s + 5·26-s − 28-s − 4·29-s − 3·31-s − 32-s + 35-s + 4·37-s + 4·38-s + 40-s − 3·41-s + 8·43-s + 44-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 + 0.301·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.980·26-s − 0.188·28-s − 0.742·29-s − 0.538·31-s − 0.176·32-s + 0.169·35-s + 0.657·37-s + 0.648·38-s + 0.158·40-s − 0.468·41-s + 1.21·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2718165095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2718165095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.05835066200811, −12.53237830283043, −12.20802567270168, −11.77216565415670, −11.13844030644572, −10.66079704892262, −10.50953929728093, −9.587151021123394, −9.392268338006857, −8.993792168476958, −8.437535866158758, −7.708821216936271, −7.498960762562741, −7.008773044573318, −6.484303492081985, −5.938175246499529, −5.374015236637172, −4.597938442541544, −4.363672273064324, −3.506466586557198, −2.975307061205913, −2.476007362591647, −1.788592608772499, −1.094337205395065, −0.1797625254162904,
0.1797625254162904, 1.094337205395065, 1.788592608772499, 2.476007362591647, 2.975307061205913, 3.506466586557198, 4.363672273064324, 4.597938442541544, 5.374015236637172, 5.938175246499529, 6.484303492081985, 7.008773044573318, 7.498960762562741, 7.708821216936271, 8.437535866158758, 8.993792168476958, 9.392268338006857, 9.587151021123394, 10.50953929728093, 10.66079704892262, 11.13844030644572, 11.77216565415670, 12.20802567270168, 12.53237830283043, 13.05835066200811
