L(s) = 1 | − 3.99i·5-s − 7.74i·7-s + 7.68·11-s + 1.42i·13-s − 22.3·17-s − 21.9·19-s − 18.1i·23-s + 9.02·25-s − 15.2i·29-s + 10.2i·31-s − 30.9·35-s − 59.1i·37-s − 41.3·41-s − 11.0·43-s + 63.6i·47-s + ⋯ |
L(s) = 1 | − 0.799i·5-s − 1.10i·7-s + 0.699·11-s + 0.109i·13-s − 1.31·17-s − 1.15·19-s − 0.789i·23-s + 0.360·25-s − 0.527i·29-s + 0.329i·31-s − 0.884·35-s − 1.59i·37-s − 1.00·41-s − 0.255·43-s + 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7052319264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7052319264\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.99iT - 25T^{2} \) |
| 7 | \( 1 + 7.74iT - 49T^{2} \) |
| 11 | \( 1 - 7.68T + 121T^{2} \) |
| 13 | \( 1 - 1.42iT - 169T^{2} \) |
| 17 | \( 1 + 22.3T + 289T^{2} \) |
| 19 | \( 1 + 21.9T + 361T^{2} \) |
| 23 | \( 1 + 18.1iT - 529T^{2} \) |
| 29 | \( 1 + 15.2iT - 841T^{2} \) |
| 31 | \( 1 - 10.2iT - 961T^{2} \) |
| 37 | \( 1 + 59.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 11.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 164.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 35.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 147.T + 9.40e3T^{2} \) |
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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
−8.726972475035649239436037569579, −8.074720126342498771683678231806, −6.89571117449341953890427109262, −6.56963953457113261328394136036, −5.32701388324362845054418116742, −4.23397739069071638395471358777, −4.08899308773457540768373441028, −2.45636774174422485230372817242, −1.27282481435830116132023301370, −0.18509257765859537000420117842,
1.74653184775907711815901396829, 2.62449936769427796470074675437, 3.57470246793922819791809740960, 4.64783505825808727116579057938, 5.59268118647339747200434914223, 6.63108778250218237963076527903, 6.79311189693865243427066239175, 8.186188167558365314752735777215, 8.749907294710317483441522148957, 9.465686555658611422594988833826