| L(s) = 1 | − 3·3-s + 16·7-s + 9·9-s − 24·11-s + 14·13-s + 18·17-s − 36·19-s − 48·21-s + 104·23-s − 27·27-s − 250·29-s + 28·31-s + 72·33-s + 54·37-s − 42·39-s + 354·41-s + 228·43-s + 408·47-s − 87·49-s − 54·51-s − 262·53-s + 108·57-s + 64·59-s + 374·61-s + 144·63-s + 300·67-s − 312·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.863·7-s + 1/3·9-s − 0.657·11-s + 0.298·13-s + 0.256·17-s − 0.434·19-s − 0.498·21-s + 0.942·23-s − 0.192·27-s − 1.60·29-s + 0.162·31-s + 0.379·33-s + 0.239·37-s − 0.172·39-s + 1.34·41-s + 0.808·43-s + 1.26·47-s − 0.253·49-s − 0.148·51-s − 0.679·53-s + 0.250·57-s + 0.141·59-s + 0.785·61-s + 0.287·63-s + 0.547·67-s − 0.544·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.837368807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837368807\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 104 T + p^{3} T^{2} \) |
| 29 | \( 1 + 250 T + p^{3} T^{2} \) |
| 31 | \( 1 - 28 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 - 354 T + p^{3} T^{2} \) |
| 43 | \( 1 - 228 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 262 T + p^{3} T^{2} \) |
| 59 | \( 1 - 64 T + p^{3} T^{2} \) |
| 61 | \( 1 - 374 T + p^{3} T^{2} \) |
| 67 | \( 1 - 300 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 73 | \( 1 + 274 T + p^{3} T^{2} \) |
| 79 | \( 1 + 788 T + p^{3} T^{2} \) |
| 83 | \( 1 + 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 786 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1086 T + p^{3} T^{2} \) |
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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
−8.609547213164729267612049334487, −7.68334268192296117634764356693, −7.25562372835582391158229018181, −6.09787634225944265971687377076, −5.50746292458443466117625478512, −4.72159906689128455851670469281, −3.93445636379198422888816571356, −2.71484325395550226457856355334, −1.67277668501471432767176565254, −0.63575549882363309514598685155,
0.63575549882363309514598685155, 1.67277668501471432767176565254, 2.71484325395550226457856355334, 3.93445636379198422888816571356, 4.72159906689128455851670469281, 5.50746292458443466117625478512, 6.09787634225944265971687377076, 7.25562372835582391158229018181, 7.68334268192296117634764356693, 8.609547213164729267612049334487
