| L(s) = 1 | − 2-s − 3-s + 4-s + 2.26·5-s + 6-s − 5.18·7-s − 8-s + 9-s − 2.26·10-s + 1.67·11-s − 12-s + 3.65·13-s + 5.18·14-s − 2.26·15-s + 16-s − 3.94·17-s − 18-s − 2.02·19-s + 2.26·20-s + 5.18·21-s − 1.67·22-s + 23-s + 24-s + 0.115·25-s − 3.65·26-s − 27-s − 5.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.01·5-s + 0.408·6-s − 1.95·7-s − 0.353·8-s + 0.333·9-s − 0.715·10-s + 0.506·11-s − 0.288·12-s + 1.01·13-s + 1.38·14-s − 0.583·15-s + 0.250·16-s − 0.955·17-s − 0.235·18-s − 0.464·19-s + 0.505·20-s + 1.13·21-s − 0.357·22-s + 0.208·23-s + 0.204·24-s + 0.0230·25-s − 0.717·26-s − 0.192·27-s − 0.979·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 5.18T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 3.65T + 13T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 0.341T + 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 0.919T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 8.06T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 - 6.19T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.402472388296662540643397749268, −6.93281404929463727274624046884, −6.54953398071201384960728743742, −6.23094522682716637445176121909, −5.42243879006788520906132104537, −4.16556889081898957804073998327, −3.26626950606811818235073410830, −2.35135395590151066045175330692, −1.22356959658963139312500943873, 0,
1.22356959658963139312500943873, 2.35135395590151066045175330692, 3.26626950606811818235073410830, 4.16556889081898957804073998327, 5.42243879006788520906132104537, 6.23094522682716637445176121909, 6.54953398071201384960728743742, 6.93281404929463727274624046884, 8.402472388296662540643397749268
