| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3·7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s − 4·13-s + 3·14-s − 15-s + 16-s + 18-s + 3·19-s − 20-s + 3·21-s + 2·22-s + 3·23-s + 24-s + 25-s − 4·26-s + 27-s + 3·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s − 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.654·21-s + 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.566·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.343519839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.343519839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.729411975430576469280266242473, −8.766783224822536560807817477421, −7.964490712075905205172258959594, −7.31900993228210696257364783505, −6.48779392536899655670573651803, −5.07540381220355357136279585916, −4.68800310637633344403981796395, −3.59513976702977904076465490605, −2.61286914184096961939985206997, −1.40159716100785087523266926724,
1.40159716100785087523266926724, 2.61286914184096961939985206997, 3.59513976702977904076465490605, 4.68800310637633344403981796395, 5.07540381220355357136279585916, 6.48779392536899655670573651803, 7.31900993228210696257364783505, 7.964490712075905205172258959594, 8.766783224822536560807817477421, 9.729411975430576469280266242473
