L(s) = 1 | − 2-s + 4-s − 3.76·5-s + 7-s − 8-s + 3.76·10-s − 5.76·11-s − 4.26·13-s − 14-s + 16-s − 4.26·17-s − 8.03·19-s − 3.76·20-s + 5.76·22-s − 23-s + 9.18·25-s + 4.26·26-s + 28-s − 2·29-s + 0.0811·31-s − 32-s + 4.26·34-s − 3.76·35-s + 0.579·37-s + 8.03·38-s + 3.76·40-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.68·5-s + 0.377·7-s − 0.353·8-s + 1.19·10-s − 1.73·11-s − 1.18·13-s − 0.267·14-s + 0.250·16-s − 1.03·17-s − 1.84·19-s − 0.842·20-s + 1.22·22-s − 0.208·23-s + 1.83·25-s + 0.836·26-s + 0.188·28-s − 0.371·29-s + 0.0145·31-s − 0.176·32-s + 0.731·34-s − 0.636·35-s + 0.0953·37-s + 1.30·38-s + 0.595·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1798756729\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1798756729\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3.76T + 5T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 8.03T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 0.0811T + 31T^{2} \) |
| 37 | \( 1 - 0.579T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−8.462405136107744907670253739791, −8.101211676651296109743993779642, −7.43351743042912827019965001268, −6.92420607074265708065228826544, −5.72101000082472692591178210218, −4.64187861588755323067633119306, −4.20113028576478347297335592812, −2.86065245503835289291886979269, −2.17790494929795614090795808471, −0.26697866411890018307002965675,
0.26697866411890018307002965675, 2.17790494929795614090795808471, 2.86065245503835289291886979269, 4.20113028576478347297335592812, 4.64187861588755323067633119306, 5.72101000082472692591178210218, 6.92420607074265708065228826544, 7.43351743042912827019965001268, 8.101211676651296109743993779642, 8.462405136107744907670253739791