| L(s) = 1 | + 2-s + 4-s + 1.58·7-s + 8-s − 1.41·11-s + 0.171·13-s + 1.58·14-s + 16-s − 17-s − 19-s − 1.41·22-s − 9.24·23-s + 0.171·26-s + 1.58·28-s + 5.82·29-s − 2.24·31-s + 32-s − 34-s + 8.48·37-s − 38-s − 4.24·41-s + 10.2·43-s − 1.41·44-s − 9.24·46-s − 4.48·49-s + 0.171·52-s + 11.4·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.599·7-s + 0.353·8-s − 0.426·11-s + 0.0475·13-s + 0.423·14-s + 0.250·16-s − 0.242·17-s − 0.229·19-s − 0.301·22-s − 1.92·23-s + 0.0336·26-s + 0.299·28-s + 1.08·29-s − 0.402·31-s + 0.176·32-s − 0.171·34-s + 1.39·37-s − 0.162·38-s − 0.662·41-s + 1.56·43-s − 0.213·44-s − 1.36·46-s − 0.640·49-s + 0.0237·52-s + 1.57·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.417655323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417655323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 9.24T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.908348515390718593603931495463, −6.97283679067666865370931078098, −6.31701654619386153804348470004, −5.62964146597898368826261347087, −5.00485273216513622461034039567, −4.20467668668607205994053930669, −3.72324330251560670835138603376, −2.51612897798450994033934438120, −2.08197217236639860443911343436, −0.803923554419488282220457854311,
0.803923554419488282220457854311, 2.08197217236639860443911343436, 2.51612897798450994033934438120, 3.72324330251560670835138603376, 4.20467668668607205994053930669, 5.00485273216513622461034039567, 5.62964146597898368826261347087, 6.31701654619386153804348470004, 6.97283679067666865370931078098, 7.908348515390718593603931495463
