L(s) = 1 | + (−0.309 − 0.951i)2-s + (2.54 + 1.85i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (0.973 − 2.99i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (2.13 + 6.57i)9-s − 0.999·10-s + (−3.22 + 0.780i)11-s − 3.14·12-s + (1.68 + 5.17i)13-s + (−0.809 − 0.587i)14-s + (2.54 − 1.85i)15-s + (0.309 − 0.951i)16-s + (−0.147 + 0.453i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.47 + 1.06i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.397 − 1.22i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (0.712 + 2.19i)9-s − 0.316·10-s + (−0.971 + 0.235i)11-s − 0.909·12-s + (0.466 + 1.43i)13-s + (−0.216 − 0.157i)14-s + (0.657 − 0.477i)15-s + (0.0772 − 0.237i)16-s + (−0.0357 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14988 + 0.555837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14988 + 0.555837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.22 - 0.780i)T \) |
good | 3 | \( 1 + (-2.54 - 1.85i)T + (0.927 + 2.85i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 5.17i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.147 - 0.453i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.78 - 3.47i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + (-5.31 + 3.86i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.99 + 9.21i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.69 + 2.68i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.04 + 1.48i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + (7.62 + 5.54i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.90 - 5.85i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.457 + 0.332i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.633 + 1.94i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 + (1.11 - 3.44i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.6 - 9.19i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.04 + 3.21i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.537 + 1.65i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 9.54T + 89T^{2} \) |
| 97 | \( 1 + (1.21 + 3.73i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.04110921064371175996165930159, −9.650290235095078880849592097712, −8.865567835673864323707810226362, −8.109904416237236028602963749201, −7.48361346833163530676742570225, −5.62026098047056052910677944742, −4.38296504587257946503507031573, −4.00973785460669921259808651213, −2.73245451453918038276834291257, −1.80917433210416348533555653302,
1.15024690866356120618426936638, 2.69394016678595224556622727628, 3.28860367599549868490776857493, 5.05764715675671221692774264029, 6.09943727281780486020209288390, 7.11030851743288954894319082374, 7.74161595859295006213010025548, 8.320846748374172740267544400628, 9.019839134772727354610782467108, 10.00953142824395094701482940417