| L(s) = 1 | − 2-s + 4-s + 3.28·5-s − 2.81·7-s − 8-s − 3.28·10-s + 5.17·11-s + 13-s + 2.81·14-s + 16-s − 0.699·17-s + 7.17·19-s + 3.28·20-s − 5.17·22-s − 6·23-s + 5.81·25-s − 26-s − 2.81·28-s − 2.22·29-s + 2.77·31-s − 32-s + 0.699·34-s − 9.24·35-s − 0.712·37-s − 7.17·38-s − 3.28·40-s − 0.823·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.47·5-s − 1.06·7-s − 0.353·8-s − 1.03·10-s + 1.56·11-s + 0.277·13-s + 0.751·14-s + 0.250·16-s − 0.169·17-s + 1.64·19-s + 0.735·20-s − 1.10·22-s − 1.25·23-s + 1.16·25-s − 0.196·26-s − 0.531·28-s − 0.412·29-s + 0.498·31-s − 0.176·32-s + 0.119·34-s − 1.56·35-s − 0.117·37-s − 1.16·38-s − 0.519·40-s − 0.128·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.711860712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.711860712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.28T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 17 | \( 1 + 0.699T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 0.712T + 37T^{2} \) |
| 41 | \( 1 + 0.823T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 - 0.445T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 + 0.222T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−9.303390347127515351792744086193, −8.648472432483771196600047388933, −7.45996110063730274622090661739, −6.64563809145041549600367418790, −6.11559722750192779300827610279, −5.50794558682088466161968144203, −4.02356867510451173199215886473, −3.07097493382868050596747750217, −1.99186720412819419350336481972, −1.01609030090818265263818988386,
1.01609030090818265263818988386, 1.99186720412819419350336481972, 3.07097493382868050596747750217, 4.02356867510451173199215886473, 5.50794558682088466161968144203, 6.11559722750192779300827610279, 6.64563809145041549600367418790, 7.45996110063730274622090661739, 8.648472432483771196600047388933, 9.303390347127515351792744086193
