| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s − 2·13-s + 16-s − 2·17-s − 4·19-s + 4·22-s − 8·23-s − 2·26-s + 2·29-s + 32-s − 2·34-s − 6·37-s − 4·38-s − 6·41-s + 4·43-s + 4·44-s − 8·46-s − 2·52-s − 10·53-s + 2·58-s + 12·59-s − 14·61-s + 64-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.852·22-s − 1.66·23-s − 0.392·26-s + 0.371·29-s + 0.176·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 0.277·52-s − 1.37·53-s + 0.262·58-s + 1.56·59-s − 1.79·61-s + 1/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.203865917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.203865917\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.38429058994031, −14.97098627593074, −14.36537657577099, −13.90615012500244, −13.58780135243846, −12.62601616440577, −12.33500291395382, −11.90698870541342, −11.21407861994564, −10.74263069736514, −10.00804636900321, −9.549145064078824, −8.802695529070538, −8.238764520408966, −7.600838216019027, −6.783061372294763, −6.437450794895559, −5.908367156349989, −4.980045425023789, −4.540031539530252, −3.782367571555721, −3.377418648862001, −2.124163036087959, −1.949750173538594, −0.6235557438272268,
0.6235557438272268, 1.949750173538594, 2.124163036087959, 3.377418648862001, 3.782367571555721, 4.540031539530252, 4.980045425023789, 5.908367156349989, 6.437450794895559, 6.783061372294763, 7.600838216019027, 8.238764520408966, 8.802695529070538, 9.549145064078824, 10.00804636900321, 10.74263069736514, 11.21407861994564, 11.90698870541342, 12.33500291395382, 12.62601616440577, 13.58780135243846, 13.90615012500244, 14.36537657577099, 14.97098627593074, 15.38429058994031
