| L(s) = 1 | − 3-s − 7-s + 9-s − 5·11-s + 2·13-s + 17-s − 4·19-s + 21-s + 7·23-s − 27-s + 6·29-s − 5·31-s + 5·33-s − 2·39-s − 10·41-s − 12·43-s − 7·47-s + 49-s − 51-s + 4·53-s + 4·57-s − 9·59-s − 3·61-s − 63-s − 11·67-s − 7·69-s + 7·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.242·17-s − 0.917·19-s + 0.218·21-s + 1.45·23-s − 0.192·27-s + 1.11·29-s − 0.898·31-s + 0.870·33-s − 0.320·39-s − 1.56·41-s − 1.82·43-s − 1.02·47-s + 1/7·49-s − 0.140·51-s + 0.549·53-s + 0.529·57-s − 1.17·59-s − 0.384·61-s − 0.125·63-s − 1.34·67-s − 0.842·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7660716057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7660716057\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−15.02899751152836, −14.53431066298960, −13.59580533029706, −13.26092799107923, −12.96043586813542, −12.31024490435050, −11.80530374441526, −11.10323957299556, −10.66350381608465, −10.30444783106097, −9.751547271786987, −8.976709237136297, −8.425385308741545, −7.973309515347904, −7.183659823837308, −6.706268516474927, −6.178926455789770, −5.453706450869195, −4.960967094355399, −4.518366849484632, −3.398746600841853, −3.122542448302539, −2.171171043389024, −1.383177667974870, −0.3417492570113006,
0.3417492570113006, 1.383177667974870, 2.171171043389024, 3.122542448302539, 3.398746600841853, 4.518366849484632, 4.960967094355399, 5.453706450869195, 6.178926455789770, 6.706268516474927, 7.183659823837308, 7.973309515347904, 8.425385308741545, 8.976709237136297, 9.751547271786987, 10.30444783106097, 10.66350381608465, 11.10323957299556, 11.80530374441526, 12.31024490435050, 12.96043586813542, 13.26092799107923, 13.59580533029706, 14.53431066298960, 15.02899751152836
