| L(s) = 1 | − 3·3-s + 19·7-s + 9·9-s + 22·11-s − 13-s + 58·17-s − 53·19-s − 57·21-s − 58·23-s − 27·27-s + 22·29-s − 35·31-s − 66·33-s + 270·37-s + 3·39-s − 468·41-s + 431·43-s + 230·47-s + 18·49-s − 174·51-s + 159·57-s + 446·59-s + 127·61-s + 171·63-s + 811·67-s + 174·69-s + 36·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.02·7-s + 1/3·9-s + 0.603·11-s − 0.0213·13-s + 0.827·17-s − 0.639·19-s − 0.592·21-s − 0.525·23-s − 0.192·27-s + 0.140·29-s − 0.202·31-s − 0.348·33-s + 1.19·37-s + 0.0123·39-s − 1.78·41-s + 1.52·43-s + 0.713·47-s + 0.0524·49-s − 0.477·51-s + 0.369·57-s + 0.984·59-s + 0.266·61-s + 0.341·63-s + 1.47·67-s + 0.303·69-s + 0.0601·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.913936669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913936669\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 53 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 35 T + p^{3} T^{2} \) |
| 37 | \( 1 - 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 468 T + p^{3} T^{2} \) |
| 43 | \( 1 - 431 T + p^{3} T^{2} \) |
| 47 | \( 1 - 230 T + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 - 446 T + p^{3} T^{2} \) |
| 61 | \( 1 - 127 T + p^{3} T^{2} \) |
| 67 | \( 1 - 811 T + p^{3} T^{2} \) |
| 71 | \( 1 - 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 522 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1368 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1138 T + p^{3} T^{2} \) |
| 89 | \( 1 - 144 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1079 T + p^{3} T^{2} \) |
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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
−10.39088651658994165538008491560, −9.468698687870373270937583522282, −8.407663077807796529324863441937, −7.64239262351513397028794959460, −6.58926865025042990638046631450, −5.65269792343558904515760488527, −4.72975676078603205047560489751, −3.76354237765496788760058019501, −2.08867865651075280038107703353, −0.887623715705447190301704751726,
0.887623715705447190301704751726, 2.08867865651075280038107703353, 3.76354237765496788760058019501, 4.72975676078603205047560489751, 5.65269792343558904515760488527, 6.58926865025042990638046631450, 7.64239262351513397028794959460, 8.407663077807796529324863441937, 9.468698687870373270937583522282, 10.39088651658994165538008491560
