| L(s) = 1 | + 3·5-s − 7-s + 5·11-s + 13-s + 17-s + 3·19-s + 5·23-s + 4·25-s + 2·29-s − 3·35-s + 2·37-s − 7·41-s + 3·43-s − 12·47-s + 49-s − 8·53-s + 15·55-s + 8·59-s + 6·61-s + 3·65-s − 8·67-s − 2·73-s − 5·77-s + 12·79-s − 10·83-s + 3·85-s − 91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 1.50·11-s + 0.277·13-s + 0.242·17-s + 0.688·19-s + 1.04·23-s + 4/5·25-s + 0.371·29-s − 0.507·35-s + 0.328·37-s − 1.09·41-s + 0.457·43-s − 1.75·47-s + 1/7·49-s − 1.09·53-s + 2.02·55-s + 1.04·59-s + 0.768·61-s + 0.372·65-s − 0.977·67-s − 0.234·73-s − 0.569·77-s + 1.35·79-s − 1.09·83-s + 0.325·85-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923019906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923019906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.660521073599250388876941205613, −7.56159279780586624950495690471, −6.60778074278395561608464853214, −6.37654696544159939234913717859, −5.49619361842659458666039074243, −4.77853450422394960525408875248, −3.68054222771730173646321769746, −2.95122974888866412245072902936, −1.80794794923846240522787037360, −1.06291647749258958169255136559,
1.06291647749258958169255136559, 1.80794794923846240522787037360, 2.95122974888866412245072902936, 3.68054222771730173646321769746, 4.77853450422394960525408875248, 5.49619361842659458666039074243, 6.37654696544159939234913717859, 6.60778074278395561608464853214, 7.56159279780586624950495690471, 8.660521073599250388876941205613
