| L(s) = 1 | + 2-s − 3.28·3-s + 4-s − 5-s − 3.28·6-s − 0.411·7-s + 8-s + 7.79·9-s − 10-s + 11-s − 3.28·12-s − 0.752·13-s − 0.411·14-s + 3.28·15-s + 16-s + 0.494·17-s + 7.79·18-s + 5.80·19-s − 20-s + 1.35·21-s + 22-s − 4.41·23-s − 3.28·24-s + 25-s − 0.752·26-s − 15.7·27-s − 0.411·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.89·3-s + 0.5·4-s − 0.447·5-s − 1.34·6-s − 0.155·7-s + 0.353·8-s + 2.59·9-s − 0.316·10-s + 0.301·11-s − 0.948·12-s − 0.208·13-s − 0.110·14-s + 0.848·15-s + 0.250·16-s + 0.119·17-s + 1.83·18-s + 1.33·19-s − 0.223·20-s + 0.295·21-s + 0.213·22-s − 0.920·23-s − 0.670·24-s + 0.200·25-s − 0.147·26-s − 3.03·27-s − 0.0777·28-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
| \(L(1)\) |
\(\approx\) |
\(1.440733723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440733723\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 + 0.411T + 7T^{2} \) |
| 13 | \( 1 + 0.752T + 13T^{2} \) |
| 17 | \( 1 - 0.494T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 - 4.08T + 37T^{2} \) |
| 41 | \( 1 + 0.339T + 41T^{2} \) |
| 43 | \( 1 - 5.67T + 43T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 + 8.25T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−7.49051040324179501024209281737, −6.90006349565284487343364609561, −6.33781818303558458248458709881, −5.57730088155965965711828017356, −5.26342504885888078554584664340, −4.30115832071376204257518747749, −3.96689160469886247876346871098, −2.80722893697846087793117400979, −1.50927856780440007610617433196, −0.62624136522844310180280943425,
0.62624136522844310180280943425, 1.50927856780440007610617433196, 2.80722893697846087793117400979, 3.96689160469886247876346871098, 4.30115832071376204257518747749, 5.26342504885888078554584664340, 5.57730088155965965711828017356, 6.33781818303558458248458709881, 6.90006349565284487343364609561, 7.49051040324179501024209281737
