| L(s) = 1 | − 2·7-s + 3·11-s + 4·13-s − 7·17-s + 4·19-s + 6·23-s − 10·29-s − 2·31-s + 2·37-s − 6·41-s − 9·43-s + 12·47-s − 3·49-s − 4·53-s − 9·59-s + 6·61-s + 4·67-s − 8·71-s − 10·73-s − 6·77-s + 12·79-s + 9·83-s − 15·89-s − 8·91-s − 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s + 1.10·13-s − 1.69·17-s + 0.917·19-s + 1.25·23-s − 1.85·29-s − 0.359·31-s + 0.328·37-s − 0.937·41-s − 1.37·43-s + 1.75·47-s − 3/7·49-s − 0.549·53-s − 1.17·59-s + 0.768·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.683·77-s + 1.35·79-s + 0.987·83-s − 1.58·89-s − 0.838·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.604006172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604006172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−12.86525880295082, −12.51023904895344, −11.73077392634633, −11.40601925845198, −11.04680795394869, −10.68050581353181, −9.897597326125194, −9.531354461190723, −9.049263647778230, −8.772447416257342, −8.342237916110056, −7.397875713242307, −7.194477309178703, −6.679149944270501, −6.125604822190417, −5.883466513374964, −5.071635997625394, −4.663164671858413, −3.922677206874973, −3.541912895886315, −3.163882283945110, −2.371545456646482, −1.703608516603336, −1.199325788899044, −0.3550916066449848,
0.3550916066449848, 1.199325788899044, 1.703608516603336, 2.371545456646482, 3.163882283945110, 3.541912895886315, 3.922677206874973, 4.663164671858413, 5.071635997625394, 5.883466513374964, 6.125604822190417, 6.679149944270501, 7.194477309178703, 7.397875713242307, 8.342237916110056, 8.772447416257342, 9.049263647778230, 9.531354461190723, 9.897597326125194, 10.68050581353181, 11.04680795394869, 11.40601925845198, 11.73077392634633, 12.51023904895344, 12.86525880295082
