L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s − 0.449·7-s + 8-s + 9-s + 4·10-s − 4.89·11-s + 12-s + 2.89·13-s − 0.449·14-s + 4·15-s + 16-s + 4.89·17-s + 18-s + 0.449·19-s + 4·20-s − 0.449·21-s − 4.89·22-s − 23-s + 24-s + 11·25-s + 2.89·26-s + 27-s − 0.449·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.78·5-s + 0.408·6-s − 0.169·7-s + 0.353·8-s + 0.333·9-s + 1.26·10-s − 1.47·11-s + 0.288·12-s + 0.804·13-s − 0.120·14-s + 1.03·15-s + 0.250·16-s + 1.18·17-s + 0.235·18-s + 0.103·19-s + 0.894·20-s − 0.0980·21-s − 1.04·22-s − 0.208·23-s + 0.204·24-s + 2.20·25-s + 0.568·26-s + 0.192·27-s − 0.0849·28-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
\(L(1)\) |
\(\approx\) |
\(5.294503506\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.294503506\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4T + 5T^{2} \) |
| 7 | \( 1 + 0.449T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 0.898T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 0.898T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.403090421328483914470569760146, −7.68474069536071076522704141009, −6.86220626190112201930814571270, −5.93677371692860366359229954731, −5.57786271831326875210536455653, −4.89548307172694561832298614343, −3.67721095474791598088477050452, −2.86364388705708469923587278650, −2.21983711838196141782436989069, −1.29017147924949527991854214699,
1.29017147924949527991854214699, 2.21983711838196141782436989069, 2.86364388705708469923587278650, 3.67721095474791598088477050452, 4.89548307172694561832298614343, 5.57786271831326875210536455653, 5.93677371692860366359229954731, 6.86220626190112201930814571270, 7.68474069536071076522704141009, 8.403090421328483914470569760146