| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 11-s + 15-s + 6·17-s + 6·19-s − 2·21-s − 5·23-s + 25-s + 27-s + 3·29-s + 7·31-s − 33-s − 2·35-s + 11·37-s + 10·41-s − 43-s + 45-s − 13·47-s − 3·49-s + 6·51-s + 10·53-s − 55-s + 6·57-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 1.45·17-s + 1.37·19-s − 0.436·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 1.25·31-s − 0.174·33-s − 0.338·35-s + 1.80·37-s + 1.56·41-s − 0.152·43-s + 0.149·45-s − 1.89·47-s − 3/7·49-s + 0.840·51-s + 1.37·53-s − 0.134·55-s + 0.794·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.231207842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.231207842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−15.83642421929307, −14.93579062441830, −14.52089284382541, −14.01679188644816, −13.50833196985862, −12.97809761589814, −12.50408370112144, −11.77191784771218, −11.42352476599087, −10.31133849543258, −9.895792350056255, −9.774279361749720, −9.012699622216016, −8.230743683771862, −7.765673799896551, −7.267905238063005, −6.352433945237409, −5.955337001595730, −5.275246559628729, −4.478340116663053, −3.731711993273592, −2.915025361815441, −2.694142474108718, −1.502925322966483, −0.7569328918158696,
0.7569328918158696, 1.502925322966483, 2.694142474108718, 2.915025361815441, 3.731711993273592, 4.478340116663053, 5.275246559628729, 5.955337001595730, 6.352433945237409, 7.267905238063005, 7.765673799896551, 8.230743683771862, 9.012699622216016, 9.774279361749720, 9.895792350056255, 10.31133849543258, 11.42352476599087, 11.77191784771218, 12.50408370112144, 12.97809761589814, 13.50833196985862, 14.01679188644816, 14.52089284382541, 14.93579062441830, 15.83642421929307
