| L(s) = 1 | + 3·3-s + 5-s − 3·7-s + 6·9-s + 6·11-s − 13-s + 3·15-s + 17-s − 9·21-s − 6·23-s − 4·25-s + 9·27-s + 10·29-s + 18·33-s − 3·35-s + 9·37-s − 3·39-s − 4·41-s + 9·43-s + 6·45-s + 9·47-s + 2·49-s + 3·51-s − 2·53-s + 6·55-s + 12·61-s − 18·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 1.13·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 0.774·15-s + 0.242·17-s − 1.96·21-s − 1.25·23-s − 4/5·25-s + 1.73·27-s + 1.85·29-s + 3.13·33-s − 0.507·35-s + 1.47·37-s − 0.480·39-s − 0.624·41-s + 1.37·43-s + 0.894·45-s + 1.31·47-s + 2/7·49-s + 0.420·51-s − 0.274·53-s + 0.809·55-s + 1.53·61-s − 2.26·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.910018050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.910018050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.716096350613733200583430676309, −8.055411431659714516662470772921, −7.17698479752612779222586183849, −6.50797373947843415101159166060, −5.86711251912006631552838499432, −4.24614402089154122391643205753, −3.91411585351240968379691466174, −2.96232476187923936667315525634, −2.27750536513965917199198575563, −1.18270614925725241154681491025,
1.18270614925725241154681491025, 2.27750536513965917199198575563, 2.96232476187923936667315525634, 3.91411585351240968379691466174, 4.24614402089154122391643205753, 5.86711251912006631552838499432, 6.50797373947843415101159166060, 7.17698479752612779222586183849, 8.055411431659714516662470772921, 8.716096350613733200583430676309
