| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 4·11-s + 4·13-s − 4·17-s − 4·19-s − 4·21-s + 6·23-s + 4·27-s − 29-s − 8·33-s + 8·37-s − 8·39-s + 10·41-s − 2·43-s + 6·47-s − 3·49-s + 8·51-s + 4·53-s + 8·57-s − 4·59-s − 2·61-s + 2·63-s − 2·67-s − 12·69-s + 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s − 0.185·29-s − 1.39·33-s + 1.31·37-s − 1.28·39-s + 1.56·41-s − 0.304·43-s + 0.875·47-s − 3/7·49-s + 1.12·51-s + 0.549·53-s + 1.05·57-s − 0.520·59-s − 0.256·61-s + 0.251·63-s − 0.244·67-s − 1.44·69-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560053468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560053468\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 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 |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.174257136877773976451954995003, −7.20846753700680099317923878653, −6.44687913621220571723080353385, −6.11917312474781851084428157080, −5.29244087237544031849181444036, −4.43716165354727915133976653549, −4.02517656701588947831444453550, −2.73970620346345026101716948672, −1.56388138400304117041442392121, −0.75927261610885969979763089229,
0.75927261610885969979763089229, 1.56388138400304117041442392121, 2.73970620346345026101716948672, 4.02517656701588947831444453550, 4.43716165354727915133976653549, 5.29244087237544031849181444036, 6.11917312474781851084428157080, 6.44687913621220571723080353385, 7.20846753700680099317923878653, 8.174257136877773976451954995003
