L(s) = 1 | − 7.62i·5-s + 2.45·7-s + 3.31i·11-s + 20.4·13-s + 14.2i·17-s + 25.2·19-s − 7.64i·23-s − 33.1·25-s + 43.4i·29-s + 33.4·31-s − 18.7i·35-s + 24.5·37-s + 21.8i·41-s + 17.2·43-s + 76.8i·47-s + ⋯ |
L(s) = 1 | − 1.52i·5-s + 0.350·7-s + 0.301i·11-s + 1.57·13-s + 0.841i·17-s + 1.32·19-s − 0.332i·23-s − 1.32·25-s + 1.49i·29-s + 1.07·31-s − 0.535i·35-s + 0.664·37-s + 0.531i·41-s + 0.401·43-s + 1.63i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.453308974\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.453308974\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
good | 5 | \( 1 + 7.62iT - 25T^{2} \) |
| 7 | \( 1 - 2.45T + 49T^{2} \) |
| 13 | \( 1 - 20.4T + 169T^{2} \) |
| 17 | \( 1 - 14.2iT - 289T^{2} \) |
| 19 | \( 1 - 25.2T + 361T^{2} \) |
| 23 | \( 1 + 7.64iT - 529T^{2} \) |
| 29 | \( 1 - 43.4iT - 841T^{2} \) |
| 31 | \( 1 - 33.4T + 961T^{2} \) |
| 37 | \( 1 - 24.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 17.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 3.42iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 30.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 131.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−9.119262017058323295428435785717, −8.275312413116930657134947245250, −7.959791161403506449274363129145, −6.62615281910753819886750522808, −5.77303048072790242849859410636, −4.96607696602853650718298055814, −4.25603313322684304391970426522, −3.22521764326752071163056318266, −1.56006321788222548119611289920, −0.981337021087652131914697979169,
0.941825803495554317333389207388, 2.40943484420917676160266559641, 3.25284768066112492628288316044, 4.03161146161981570617124449216, 5.37705387462866729788999606278, 6.16131908687979466986666088595, 6.87202484108601885986548645325, 7.67619854396990497896858392120, 8.378101028595195956327318827027, 9.450352509375509580585870409782