| L(s) = 1 | + 3·3-s − 5·5-s + 12·7-s + 9·9-s − 24·11-s − 38·13-s − 15·15-s − 6·17-s + 104·19-s + 36·21-s − 100·23-s + 25·25-s + 27·27-s − 230·29-s + 56·31-s − 72·33-s − 60·35-s − 190·37-s − 114·39-s + 202·41-s − 148·43-s − 45·45-s − 124·47-s − 199·49-s − 18·51-s − 206·53-s + 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.647·7-s + 1/3·9-s − 0.657·11-s − 0.810·13-s − 0.258·15-s − 0.0856·17-s + 1.25·19-s + 0.374·21-s − 0.906·23-s + 1/5·25-s + 0.192·27-s − 1.47·29-s + 0.324·31-s − 0.379·33-s − 0.289·35-s − 0.844·37-s − 0.468·39-s + 0.769·41-s − 0.524·43-s − 0.149·45-s − 0.384·47-s − 0.580·49-s − 0.0494·51-s − 0.533·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 190 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 124 T + p^{3} T^{2} \) |
| 53 | \( 1 + 206 T + p^{3} T^{2} \) |
| 59 | \( 1 + 128 T + p^{3} T^{2} \) |
| 61 | \( 1 + 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 204 T + p^{3} T^{2} \) |
| 71 | \( 1 - 440 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 79 | \( 1 + 816 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1412 T + p^{3} T^{2} \) |
| 89 | \( 1 + 214 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1202 T + p^{3} T^{2} \) |
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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
−9.282296051467975522125667996070, −8.142363929842809234706105463883, −7.75684730155721592930073376064, −6.94980187619923034078811211794, −5.56385271295564639586285409180, −4.79146062091131050526406040296, −3.74610531081243546386164291763, −2.71379767898427593977972690361, −1.60236973578665625602832087740, 0,
1.60236973578665625602832087740, 2.71379767898427593977972690361, 3.74610531081243546386164291763, 4.79146062091131050526406040296, 5.56385271295564639586285409180, 6.94980187619923034078811211794, 7.75684730155721592930073376064, 8.142363929842809234706105463883, 9.282296051467975522125667996070
