| L(s) = 1 | + 5-s − 3·7-s + 5·13-s − 3·17-s + 23-s − 4·25-s − 3·29-s + 5·31-s − 3·35-s + 3·37-s − 10·41-s + 6·43-s + 3·47-s + 2·49-s + 10·53-s − 8·59-s + 8·61-s + 5·65-s + 11·67-s + 3·71-s + 6·73-s + 79-s + 6·83-s − 3·85-s + 16·89-s − 15·91-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 1.38·13-s − 0.727·17-s + 0.208·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s − 0.507·35-s + 0.493·37-s − 1.56·41-s + 0.914·43-s + 0.437·47-s + 2/7·49-s + 1.37·53-s − 1.04·59-s + 1.02·61-s + 0.620·65-s + 1.34·67-s + 0.356·71-s + 0.702·73-s + 0.112·79-s + 0.658·83-s − 0.325·85-s + 1.69·89-s − 1.57·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.468251260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.468251260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−13.13051349781459, −12.79240228006369, −12.10044743382050, −11.66134253177173, −11.19074515281587, −10.62895106069463, −10.27256125284384, −9.731286488383858, −9.254382349496506, −8.962530261756539, −8.289603966737793, −7.961562689676261, −7.146353487814487, −6.684812231778760, −6.341030840966859, −5.848548383088903, −5.442934484262600, −4.680423917455344, −4.082104214576656, −3.539504959524621, −3.209963550372227, −2.318396488242236, −2.001976479323219, −1.051883100317590, −0.4946625880526404,
0.4946625880526404, 1.051883100317590, 2.001976479323219, 2.318396488242236, 3.209963550372227, 3.539504959524621, 4.082104214576656, 4.680423917455344, 5.442934484262600, 5.848548383088903, 6.341030840966859, 6.684812231778760, 7.146353487814487, 7.961562689676261, 8.289603966737793, 8.962530261756539, 9.254382349496506, 9.731286488383858, 10.27256125284384, 10.62895106069463, 11.19074515281587, 11.66134253177173, 12.10044743382050, 12.79240228006369, 13.13051349781459
