| L(s) = 1 | − 3·3-s − 5·5-s + 4·7-s + 9·9-s + 72·11-s − 6·13-s + 15·15-s + 38·17-s + 52·19-s − 12·21-s + 152·23-s + 25·25-s − 27·27-s − 78·29-s + 120·31-s − 216·33-s − 20·35-s − 150·37-s + 18·39-s + 362·41-s − 484·43-s − 45·45-s + 280·47-s − 327·49-s − 114·51-s − 670·53-s − 360·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.215·7-s + 1/3·9-s + 1.97·11-s − 0.128·13-s + 0.258·15-s + 0.542·17-s + 0.627·19-s − 0.124·21-s + 1.37·23-s + 1/5·25-s − 0.192·27-s − 0.499·29-s + 0.695·31-s − 1.13·33-s − 0.0965·35-s − 0.666·37-s + 0.0739·39-s + 1.37·41-s − 1.71·43-s − 0.149·45-s + 0.868·47-s − 0.953·49-s − 0.313·51-s − 1.73·53-s − 0.882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.399546814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.399546814\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 120 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 670 T + p^{3} T^{2} \) |
| 59 | \( 1 - 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 222 T + p^{3} T^{2} \) |
| 67 | \( 1 + 4 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 994 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1634 T + p^{3} T^{2} \) |
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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
−12.78240739006483210695379057508, −11.77549896986239482811044469711, −11.25768762133576898249486631491, −9.832276672292791059282854152810, −8.806986510515434571512739397121, −7.36329013318850519948758778500, −6.36727792563576377964668038282, −4.91426245679292978087357297984, −3.59203794348390889867775206923, −1.16190553529741305767696046865,
1.16190553529741305767696046865, 3.59203794348390889867775206923, 4.91426245679292978087357297984, 6.36727792563576377964668038282, 7.36329013318850519948758778500, 8.806986510515434571512739397121, 9.832276672292791059282854152810, 11.25768762133576898249486631491, 11.77549896986239482811044469711, 12.78240739006483210695379057508
