| L(s) = 1 | − 5·2-s + 17·4-s − 7·5-s + 13·7-s − 45·8-s + 35·10-s − 26·11-s − 65·14-s + 89·16-s − 77·17-s + 126·19-s − 119·20-s + 130·22-s + 96·23-s − 76·25-s + 221·28-s + 82·29-s − 196·31-s − 85·32-s + 385·34-s − 91·35-s + 131·37-s − 630·38-s + 315·40-s + 336·41-s − 201·43-s − 442·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 0.626·5-s + 0.701·7-s − 1.98·8-s + 1.10·10-s − 0.712·11-s − 1.24·14-s + 1.39·16-s − 1.09·17-s + 1.52·19-s − 1.33·20-s + 1.25·22-s + 0.870·23-s − 0.607·25-s + 1.49·28-s + 0.525·29-s − 1.13·31-s − 0.469·32-s + 1.94·34-s − 0.439·35-s + 0.582·37-s − 2.68·38-s + 1.24·40-s + 1.27·41-s − 0.712·43-s − 1.51·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 - 126 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 - 131 T + p^{3} T^{2} \) |
| 41 | \( 1 - 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 201 T + p^{3} T^{2} \) |
| 47 | \( 1 + 105 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 + 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 + 308 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−8.734760012427977889802254294901, −7.87180965623879135931531860106, −7.54326851861019013554170146958, −6.73478700623421280187707382153, −5.55376550235472465605066509424, −4.51570671946648497896695462422, −3.12934645262366110467652209239, −2.10925236850159177021432798024, −1.03745381495586958894139564729, 0,
1.03745381495586958894139564729, 2.10925236850159177021432798024, 3.12934645262366110467652209239, 4.51570671946648497896695462422, 5.55376550235472465605066509424, 6.73478700623421280187707382153, 7.54326851861019013554170146958, 7.87180965623879135931531860106, 8.734760012427977889802254294901
