| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 4·11-s + 4·13-s + 15-s + 17-s − 4·19-s + 2·21-s + 4·23-s + 25-s + 27-s − 10·29-s + 4·33-s + 2·35-s + 2·37-s + 4·39-s + 4·41-s − 10·43-s + 45-s − 3·49-s + 51-s − 2·53-s + 4·55-s − 4·57-s + 2·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.624·41-s − 1.52·43-s + 0.149·45-s − 3/7·49-s + 0.140·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.935090342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.935090342\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−9.051847317551521100613257277892, −8.497753703559820135349786623129, −7.69617674082702102669180579151, −6.74619398694817837843337274866, −6.07522311121004303923167697519, −5.07485777240852541322558549872, −4.09089498903595190596807990628, −3.38770868820964526555865969320, −2.03549912634207135657753200213, −1.28213981068191233640030972352,
1.28213981068191233640030972352, 2.03549912634207135657753200213, 3.38770868820964526555865969320, 4.09089498903595190596807990628, 5.07485777240852541322558549872, 6.07522311121004303923167697519, 6.74619398694817837843337274866, 7.69617674082702102669180579151, 8.497753703559820135349786623129, 9.051847317551521100613257277892
