L(s) = 1 | + (1.15 + 0.814i)2-s + (0.887 − 1.48i)3-s + (0.672 + 1.88i)4-s + (2.23 − 0.995i)6-s + 0.797·7-s + (−0.757 + 2.72i)8-s + (−1.42 − 2.64i)9-s + 0.320i·11-s + (3.39 + 0.672i)12-s + 4.30·13-s + (0.921 + 0.649i)14-s + (−3.09 + 2.53i)16-s + 2.57·17-s + (0.506 − 4.21i)18-s + 6.10·19-s + ⋯ |
L(s) = 1 | + (0.817 + 0.576i)2-s + (0.512 − 0.858i)3-s + (0.336 + 0.941i)4-s + (0.913 − 0.406i)6-s + 0.301·7-s + (−0.267 + 0.963i)8-s + (−0.474 − 0.880i)9-s + 0.0966i·11-s + (0.980 + 0.194i)12-s + 1.19·13-s + (0.246 + 0.173i)14-s + (−0.773 + 0.633i)16-s + 0.624·17-s + (0.119 − 0.992i)18-s + 1.40·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.82026 + 0.423132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.82026 + 0.423132i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 - 0.814i)T \) |
| 3 | \( 1 + (-0.887 + 1.48i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.797T + 7T^{2} \) |
| 11 | \( 1 - 0.320iT - 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 - 3.13iT - 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 31 | \( 1 + 9.90iT - 31T^{2} \) |
| 37 | \( 1 + 8.49T + 37T^{2} \) |
| 41 | \( 1 + 5.28iT - 41T^{2} \) |
| 43 | \( 1 - 2.97iT - 43T^{2} \) |
| 47 | \( 1 - 6.56iT - 47T^{2} \) |
| 53 | \( 1 - 3.94iT - 53T^{2} \) |
| 59 | \( 1 - 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 8.83iT - 61T^{2} \) |
| 67 | \( 1 - 4.66iT - 67T^{2} \) |
| 71 | \( 1 + 3.43T + 71T^{2} \) |
| 73 | \( 1 - 1.43iT - 73T^{2} \) |
| 79 | \( 1 + 2.89iT - 79T^{2} \) |
| 83 | \( 1 + 3.37T + 83T^{2} \) |
| 89 | \( 1 + 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 4.26iT - 97T^{2} \) |
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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
−11.17012749706276052546037422500, −9.549922209401546033162968418504, −8.674469804123384572657535063372, −7.70114093714813459308229757721, −7.31811069647351522501062620602, −6.06678074982199070036652615055, −5.48094762741343895142853221027, −3.92749918352712565792187266833, −3.11643620740984926377375013883, −1.64355104061000545083358813022,
1.62117513450111748204195663702, 3.18304935562094515935866065276, 3.72892955102154574269938764363, 5.00315267699943226513811998594, 5.59051960703727345931527051999, 6.91007000239851425158101693502, 8.167626421625009530327328694039, 9.089537507503596032870687699616, 9.934518995644546280672993747788, 10.72951726261763725440123768235