L(s) = 1 | − 0.0551·3-s + 0.393·5-s − 3.87·7-s − 2.99·9-s − 1.57·11-s − 5.66·13-s − 0.0216·15-s − 7.20·17-s + 1.16·19-s + 0.213·21-s + 6.02·23-s − 4.84·25-s + 0.330·27-s − 2.81·29-s − 7.23·31-s + 0.0866·33-s − 1.52·35-s − 8.36·37-s + 0.312·39-s + 2.39·41-s + 11.9·43-s − 1.17·45-s − 3.86·47-s + 8.01·49-s + 0.397·51-s + 5.56·53-s − 0.617·55-s + ⋯ |
L(s) = 1 | − 0.0318·3-s + 0.175·5-s − 1.46·7-s − 0.998·9-s − 0.473·11-s − 1.57·13-s − 0.00559·15-s − 1.74·17-s + 0.268·19-s + 0.0466·21-s + 1.25·23-s − 0.969·25-s + 0.0636·27-s − 0.523·29-s − 1.29·31-s + 0.0150·33-s − 0.257·35-s − 1.37·37-s + 0.0499·39-s + 0.374·41-s + 1.82·43-s − 0.175·45-s − 0.563·47-s + 1.14·49-s + 0.0556·51-s + 0.765·53-s − 0.0832·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1644193011\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1644193011\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 0.0551T + 3T^{2} \) |
| 5 | \( 1 - 0.393T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 7.20T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 8.36T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.86T + 47T^{2} \) |
| 53 | \( 1 - 5.56T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 + 6.38T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.290T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.53526458410609871483979570440, −7.24061424362600513528012279908, −6.41489116800525151983569687911, −5.78841927661250394799337889394, −5.13648284693297787360545355983, −4.31673266327500783475820847061, −3.29226941968015437201522537491, −2.72873475923667287572781721817, −2.03254661219703164730904554970, −0.18181663869429397696735280708,
0.18181663869429397696735280708, 2.03254661219703164730904554970, 2.72873475923667287572781721817, 3.29226941968015437201522537491, 4.31673266327500783475820847061, 5.13648284693297787360545355983, 5.78841927661250394799337889394, 6.41489116800525151983569687911, 7.24061424362600513528012279908, 7.53526458410609871483979570440