| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·13-s − 6·17-s + 19-s + 2·21-s − 6·23-s − 27-s + 6·29-s + 8·31-s + 4·37-s − 4·39-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 6·51-s − 12·53-s − 57-s + 12·59-s − 10·61-s − 2·63-s + 4·67-s + 6·69-s − 14·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s + 0.229·19-s + 0.436·21-s − 1.25·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 0.840·51-s − 1.64·53-s − 0.132·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 1.63·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.86201280320023629368268323452, −6.67380974090865637446420647887, −6.44326962312260980200343913920, −5.84213731011766911038320732004, −4.76026404760386656591663844528, −4.18984691243596275122350721193, −3.29346001739995212283696395399, −2.35602169951602417312881218236, −1.18851669072197634226947877736, 0,
1.18851669072197634226947877736, 2.35602169951602417312881218236, 3.29346001739995212283696395399, 4.18984691243596275122350721193, 4.76026404760386656591663844528, 5.84213731011766911038320732004, 6.44326962312260980200343913920, 6.67380974090865637446420647887, 7.86201280320023629368268323452
