| L(s) = 1 | + 2.83·3-s − 0.941·5-s + 0.892·7-s + 5.03·9-s + 4.24·11-s + 5.92·13-s − 2.66·15-s − 3.63·17-s + 1.37·19-s + 2.52·21-s + 6.93·23-s − 4.11·25-s + 5.76·27-s + 6.43·29-s − 5.66·31-s + 12.0·33-s − 0.840·35-s + 9.88·37-s + 16.8·39-s − 0.298·41-s + 5.80·43-s − 4.74·45-s − 3.60·47-s − 6.20·49-s − 10.3·51-s + 1.80·53-s − 4.00·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.421·5-s + 0.337·7-s + 1.67·9-s + 1.28·11-s + 1.64·13-s − 0.689·15-s − 0.882·17-s + 0.316·19-s + 0.552·21-s + 1.44·23-s − 0.822·25-s + 1.10·27-s + 1.19·29-s − 1.01·31-s + 2.09·33-s − 0.142·35-s + 1.62·37-s + 2.69·39-s − 0.0465·41-s + 0.884·43-s − 0.706·45-s − 0.525·47-s − 0.886·49-s − 1.44·51-s + 0.248·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.726321668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.726321668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.83T + 3T^{2} \) |
| 5 | \( 1 + 0.941T + 5T^{2} \) |
| 7 | \( 1 - 0.892T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 9.88T + 37T^{2} \) |
| 41 | \( 1 + 0.298T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 1.66T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 7.74T + 97T^{2} \) |
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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
−7.909006800985199678087303008298, −7.37853434299608483342366074180, −6.56101105776741268912204688494, −5.96383610780973500863432308845, −4.60626655188968360234801951107, −4.14307966392620129421539728588, −3.44242596576110014590397557647, −2.86405133718482924394442803647, −1.75770730077733034613810654068, −1.11392708171217193826259378033,
1.11392708171217193826259378033, 1.75770730077733034613810654068, 2.86405133718482924394442803647, 3.44242596576110014590397557647, 4.14307966392620129421539728588, 4.60626655188968360234801951107, 5.96383610780973500863432308845, 6.56101105776741268912204688494, 7.37853434299608483342366074180, 7.909006800985199678087303008298
