| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s − 13·7-s + 8·8-s − 10·10-s + 15·11-s + 13·13-s − 26·14-s + 16·16-s + 75·17-s − 130·19-s − 20·20-s + 30·22-s − 45·23-s + 25·25-s + 26·26-s − 52·28-s + 138·29-s − 34·31-s + 32·32-s + 150·34-s + 65·35-s − 379·37-s − 260·38-s − 40·40-s − 243·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.701·7-s + 0.353·8-s − 0.316·10-s + 0.411·11-s + 0.277·13-s − 0.496·14-s + 1/4·16-s + 1.07·17-s − 1.56·19-s − 0.223·20-s + 0.290·22-s − 0.407·23-s + 1/5·25-s + 0.196·26-s − 0.350·28-s + 0.883·29-s − 0.196·31-s + 0.176·32-s + 0.756·34-s + 0.313·35-s − 1.68·37-s − 1.10·38-s − 0.158·40-s − 0.925·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 17 | \( 1 - 75 T + p^{3} T^{2} \) |
| 19 | \( 1 + 130 T + p^{3} T^{2} \) |
| 23 | \( 1 + 45 T + p^{3} T^{2} \) |
| 29 | \( 1 - 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 34 T + p^{3} T^{2} \) |
| 37 | \( 1 + 379 T + p^{3} T^{2} \) |
| 41 | \( 1 + 243 T + p^{3} T^{2} \) |
| 43 | \( 1 - 416 T + p^{3} T^{2} \) |
| 47 | \( 1 + 378 T + p^{3} T^{2} \) |
| 53 | \( 1 - 3 T + p^{3} T^{2} \) |
| 59 | \( 1 - 816 T + p^{3} T^{2} \) |
| 61 | \( 1 + 607 T + p^{3} T^{2} \) |
| 67 | \( 1 + 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 57 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1162 T + p^{3} T^{2} \) |
| 79 | \( 1 + T + p^{3} T^{2} \) |
| 83 | \( 1 + 672 T + p^{3} T^{2} \) |
| 89 | \( 1 + 969 T + p^{3} T^{2} \) |
| 97 | \( 1 + 949 T + p^{3} T^{2} \) |
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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
−8.881747179531307960901790433017, −8.162637367752187941781066336655, −7.11220817825591783891104594162, −6.43195554483633943337836645558, −5.63942151657470634623985011471, −4.51203523263413404402633365729, −3.72025803666513258443009479409, −2.90046869167590825300862472619, −1.54295737085065862995283325377, 0,
1.54295737085065862995283325377, 2.90046869167590825300862472619, 3.72025803666513258443009479409, 4.51203523263413404402633365729, 5.63942151657470634623985011471, 6.43195554483633943337836645558, 7.11220817825591783891104594162, 8.162637367752187941781066336655, 8.881747179531307960901790433017
