| L(s) = 1 | + 2-s + 4-s + 3.12·7-s + 8-s − 2·11-s − 4·13-s + 3.12·14-s + 16-s + 3.12·17-s − 19-s − 2·22-s − 4·26-s + 3.12·28-s + 2·29-s + 9.12·31-s + 32-s + 3.12·34-s − 38-s + 5.12·41-s − 10.2·43-s − 2·44-s + 10.2·47-s + 2.75·49-s − 4·52-s − 4.24·53-s + 3.12·56-s + 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.18·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s + 0.834·14-s + 0.250·16-s + 0.757·17-s − 0.229·19-s − 0.426·22-s − 0.784·26-s + 0.590·28-s + 0.371·29-s + 1.63·31-s + 0.176·32-s + 0.535·34-s − 0.162·38-s + 0.800·41-s − 1.56·43-s − 0.301·44-s + 1.49·47-s + 0.393·49-s − 0.554·52-s − 0.583·53-s + 0.417·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.762136121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.762136121\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.73837325477904936366487296729, −7.16586161619906649071779501018, −6.28671945967212139734280274152, −5.55232640660346326548028305018, −4.80119892259496669178453973175, −4.60646526489557830519159772737, −3.49867773708386344326140631004, −2.62994011447471534576446133261, −1.99077895197642229753284680654, −0.859905326152212030710267479814,
0.859905326152212030710267479814, 1.99077895197642229753284680654, 2.62994011447471534576446133261, 3.49867773708386344326140631004, 4.60646526489557830519159772737, 4.80119892259496669178453973175, 5.55232640660346326548028305018, 6.28671945967212139734280274152, 7.16586161619906649071779501018, 7.73837325477904936366487296729
