| L(s) = 1 | − 3-s + 9-s − 11-s − 2·13-s − 6·17-s + 4·23-s − 27-s + 2·29-s + 33-s + 10·37-s + 2·39-s + 6·41-s − 8·43-s − 4·47-s − 7·49-s + 6·51-s + 6·53-s + 12·59-s + 2·61-s + 4·67-s − 4·69-s − 12·71-s + 14·73-s − 16·79-s + 81-s − 12·83-s − 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.834·23-s − 0.192·27-s + 0.371·29-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.583·47-s − 49-s + 0.840·51-s + 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.481·69-s − 1.42·71-s + 1.63·73-s − 1.80·79-s + 1/9·81-s − 1.31·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−16.44173197591392, −16.05703499579419, −15.42942837974958, −14.85081472549784, −14.42857671584835, −13.51141307827266, −13.00078655463338, −12.76923709611826, −11.82714748098699, −11.32329173991522, −11.02453419183341, −10.10131682377570, −9.811852042412011, −8.942763900355690, −8.454029019010479, −7.625350240384435, −7.032946174812028, −6.461499998621302, −5.837645896458972, −4.951183652944806, −4.631056628397896, −3.787201347757894, −2.775068453709845, −2.151489883267947, −1.020626861059094, 0,
1.020626861059094, 2.151489883267947, 2.775068453709845, 3.787201347757894, 4.631056628397896, 4.951183652944806, 5.837645896458972, 6.461499998621302, 7.032946174812028, 7.625350240384435, 8.454029019010479, 8.942763900355690, 9.811852042412011, 10.10131682377570, 11.02453419183341, 11.32329173991522, 11.82714748098699, 12.76923709611826, 13.00078655463338, 13.51141307827266, 14.42857671584835, 14.85081472549784, 15.42942837974958, 16.05703499579419, 16.44173197591392
