| L(s) = 1 | − 3·3-s − 8·4-s − 5·5-s + 7·7-s + 9·9-s − 11·11-s + 24·12-s − 22·13-s + 15·15-s + 64·16-s + 18·17-s − 112·19-s + 40·20-s − 21·21-s + 54·23-s + 25·25-s − 27·27-s − 56·28-s + 6·29-s + 230·31-s + 33·33-s − 35·35-s − 72·36-s + 122·37-s + 66·39-s + 336·41-s + 104·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s − 0.469·13-s + 0.258·15-s + 16-s + 0.256·17-s − 1.35·19-s + 0.447·20-s − 0.218·21-s + 0.489·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.0384·29-s + 1.33·31-s + 0.174·33-s − 0.169·35-s − 1/3·36-s + 0.542·37-s + 0.270·39-s + 1.27·41-s + 0.368·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 112 T + p^{3} T^{2} \) |
| 23 | \( 1 - 54 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 122 T + p^{3} T^{2} \) |
| 41 | \( 1 - 336 T + p^{3} T^{2} \) |
| 43 | \( 1 - 104 T + p^{3} T^{2} \) |
| 47 | \( 1 - 180 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 61 | \( 1 - 200 T + p^{3} T^{2} \) |
| 67 | \( 1 + 610 T + p^{3} T^{2} \) |
| 71 | \( 1 + 336 T + p^{3} T^{2} \) |
| 73 | \( 1 - 326 T + p^{3} T^{2} \) |
| 79 | \( 1 - 320 T + p^{3} T^{2} \) |
| 83 | \( 1 + 84 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1482 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1582 T + p^{3} T^{2} \) |
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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
−8.968291302008750365236260306844, −8.195654827978500349424131761979, −7.50322636749693055396185569266, −6.39296683035511011412032601924, −5.45387325310098349459664547114, −4.60503874036448627770779704997, −4.06037485896988892554647601673, −2.65436244643452139070570928237, −1.04829390756145543871928664789, 0,
1.04829390756145543871928664789, 2.65436244643452139070570928237, 4.06037485896988892554647601673, 4.60503874036448627770779704997, 5.45387325310098349459664547114, 6.39296683035511011412032601924, 7.50322636749693055396185569266, 8.195654827978500349424131761979, 8.968291302008750365236260306844
