L(s) = 1 | + (−1.67 − 0.448i)3-s + 0.896·5-s − i·7-s + (2.59 + 1.50i)9-s + 1.84i·13-s + (−1.50 − 0.402i)15-s − 4.12i·17-s + 0.654·19-s + (−0.448 + 1.67i)21-s + 7.12·23-s − 4.19·25-s + (−3.67 − 3.67i)27-s + 7.79·29-s − 0.896i·35-s − 10.6i·37-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)3-s + 0.401·5-s − 0.377i·7-s + (0.865 + 0.500i)9-s + 0.512i·13-s + (−0.387 − 0.103i)15-s − 1.00i·17-s + 0.150·19-s + (−0.0978 + 0.365i)21-s + 1.48·23-s − 0.839·25-s + (−0.706 − 0.707i)27-s + 1.44·29-s − 0.151i·35-s − 1.75i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06524 - 0.463715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06524 - 0.463715i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 + 0.448i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 0.896T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.84iT - 13T^{2} \) |
| 17 | \( 1 + 4.12iT - 17T^{2} \) |
| 19 | \( 1 - 0.654T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 9.59T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 6.89iT - 59T^{2} \) |
| 61 | \( 1 - 4.54iT - 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−10.52504667229631708530286399748, −9.659280278943825041539872853416, −8.788708744529045776646838462787, −7.36683798951083785508846641470, −6.95616673740756664060509744873, −5.83867403176755302798954268403, −5.05926283687006570173788630001, −4.02408298791847760887464944465, −2.36510530856770235044761467144, −0.844732997191226476839925604834,
1.24635816757981625207078040602, 2.93999047983456643756400939247, 4.32584659545503499749520719630, 5.27106114842114213317219144620, 6.06513175530223263344677867586, 6.82630180901709123196911390379, 8.031357383169704931816962497345, 9.014705910419124456379531354547, 10.00418767757280710199840317327, 10.51497134778931937830876379302