L(s) = 1 | + 2-s − 2.60·3-s + 4-s − 5-s − 2.60·6-s − 1.48·7-s + 8-s + 3.80·9-s − 10-s − 11-s − 2.60·12-s − 4.53·13-s − 1.48·14-s + 2.60·15-s + 16-s − 0.996·17-s + 3.80·18-s + 1.04·19-s − 20-s + 3.88·21-s − 22-s − 3.96·23-s − 2.60·24-s + 25-s − 4.53·26-s − 2.11·27-s − 1.48·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.447·5-s − 1.06·6-s − 0.562·7-s + 0.353·8-s + 1.26·9-s − 0.316·10-s − 0.301·11-s − 0.753·12-s − 1.25·13-s − 0.397·14-s + 0.673·15-s + 0.250·16-s − 0.241·17-s + 0.897·18-s + 0.239·19-s − 0.223·20-s + 0.847·21-s − 0.213·22-s − 0.827·23-s − 0.532·24-s + 0.200·25-s − 0.889·26-s − 0.406·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6217889445\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6217889445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 0.996T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 + 0.691T + 29T^{2} \) |
| 31 | \( 1 - 0.640T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 2.81T + 71T^{2} \) |
| 79 | \( 1 + 0.777T + 79T^{2} \) |
| 83 | \( 1 + 5.36T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.62826922287338721853318108687, −6.70101169878366684378261804738, −6.54990003570219473326025440468, −5.62006300662238667908183588668, −5.04182500517764599200665898473, −4.58074329704905243071459537817, −3.66694256736172273911143323684, −2.82913587909112079750103438597, −1.75315335438492154969033105289, −0.36460100604318261574966309867,
0.36460100604318261574966309867, 1.75315335438492154969033105289, 2.82913587909112079750103438597, 3.66694256736172273911143323684, 4.58074329704905243071459537817, 5.04182500517764599200665898473, 5.62006300662238667908183588668, 6.54990003570219473326025440468, 6.70101169878366684378261804738, 7.62826922287338721853318108687