| L(s) = 1 | + 3·3-s + 3·5-s − 3·7-s + 6·9-s + 6·11-s + 6·13-s + 9·15-s − 2·17-s − 19-s − 9·21-s − 8·23-s + 4·25-s + 9·27-s + 29-s + 2·31-s + 18·33-s − 9·35-s + 4·37-s + 18·39-s − 41-s − 10·43-s + 18·45-s − 6·47-s + 2·49-s − 6·51-s − 5·53-s + 18·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 1.13·7-s + 2·9-s + 1.80·11-s + 1.66·13-s + 2.32·15-s − 0.485·17-s − 0.229·19-s − 1.96·21-s − 1.66·23-s + 4/5·25-s + 1.73·27-s + 0.185·29-s + 0.359·31-s + 3.13·33-s − 1.52·35-s + 0.657·37-s + 2.88·39-s − 0.156·41-s − 1.52·43-s + 2.68·45-s − 0.875·47-s + 2/7·49-s − 0.840·51-s − 0.686·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.864458555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.864458555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 59 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−8.599791456493754434474086542753, −8.139942568435135876416009527981, −6.72136610741303542487376695263, −6.51534276590235620522788710540, −5.84476510638913664881514984210, −4.26553399618190477828297127438, −3.70847715428605487295086359244, −3.03150756258435225512959129754, −1.97635636450260983961790313324, −1.40590255916755611764878380648,
1.40590255916755611764878380648, 1.97635636450260983961790313324, 3.03150756258435225512959129754, 3.70847715428605487295086359244, 4.26553399618190477828297127438, 5.84476510638913664881514984210, 6.51534276590235620522788710540, 6.72136610741303542487376695263, 8.139942568435135876416009527981, 8.599791456493754434474086542753
