| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 5·13-s + 15-s − 7·17-s + 6·19-s − 21-s + 4·23-s − 4·25-s + 27-s + 9·29-s + 2·31-s − 35-s + 9·37-s + 5·39-s − 7·41-s + 6·43-s + 45-s − 2·47-s + 49-s − 7·51-s + 3·53-s + 6·57-s − 2·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.38·13-s + 0.258·15-s − 1.69·17-s + 1.37·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s + 1.67·29-s + 0.359·31-s − 0.169·35-s + 1.47·37-s + 0.800·39-s − 1.09·41-s + 0.914·43-s + 0.149·45-s − 0.291·47-s + 1/7·49-s − 0.980·51-s + 0.412·53-s + 0.794·57-s − 0.260·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.802008001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.802008001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
| 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
−14.73206665273298, −14.10617849851054, −13.56678322976717, −13.38695632188502, −12.99103165251775, −12.15025535433101, −11.58213901198582, −11.13950357966799, −10.49020426577242, −9.976764658183087, −9.422946869263134, −8.857753288727055, −8.580351570352612, −7.845190193769502, −7.239003453348615, −6.537692969373596, −6.224944745417559, −5.544722826987725, −4.703348056707894, −4.242630329835159, −3.442318060825744, −2.906282540124470, −2.276347853054972, −1.407653007570483, −0.7264081504946583,
0.7264081504946583, 1.407653007570483, 2.276347853054972, 2.906282540124470, 3.442318060825744, 4.242630329835159, 4.703348056707894, 5.544722826987725, 6.224944745417559, 6.537692969373596, 7.239003453348615, 7.845190193769502, 8.580351570352612, 8.857753288727055, 9.422946869263134, 9.976764658183087, 10.49020426577242, 11.13950357966799, 11.58213901198582, 12.15025535433101, 12.99103165251775, 13.38695632188502, 13.56678322976717, 14.10617849851054, 14.73206665273298
