| L(s) = 1 | − 2.94·3-s − 4.09·5-s − 3.85·7-s + 5.67·9-s + 2.39·11-s − 0.856·13-s + 12.0·15-s + 8.13·17-s − 2.25·19-s + 11.3·21-s + 5.62·23-s + 11.7·25-s − 7.88·27-s + 3.94·29-s − 8.39·31-s − 7.06·33-s + 15.7·35-s − 4.14·37-s + 2.52·39-s + 5.07·41-s − 8.45·43-s − 23.2·45-s + 6.63·47-s + 7.85·49-s − 23.9·51-s + 2.68·53-s − 9.81·55-s + ⋯ |
| L(s) = 1 | − 1.70·3-s − 1.83·5-s − 1.45·7-s + 1.89·9-s + 0.722·11-s − 0.237·13-s + 3.11·15-s + 1.97·17-s − 0.517·19-s + 2.47·21-s + 1.17·23-s + 2.35·25-s − 1.51·27-s + 0.732·29-s − 1.50·31-s − 1.22·33-s + 2.66·35-s − 0.682·37-s + 0.403·39-s + 0.792·41-s − 1.29·43-s − 3.46·45-s + 0.967·47-s + 1.12·49-s − 3.35·51-s + 0.368·53-s − 1.32·55-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
| \(L(1)\) |
\(\approx\) |
\(0.4014702742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4014702742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.94T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 + 0.856T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 8.39T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 + 2.71T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 1.19T + 97T^{2} \) |
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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
−7.40657053790604045136986154470, −7.12855129968799656165261561335, −6.52219115723883481619637247478, −5.76441502363501353611693013313, −5.10130315669302792826060765507, −4.28393317828554039360291719293, −3.60957613195605337441781942262, −3.09623664988971903387697038575, −1.16335135005503495850480861604, −0.41111293070407604356287744136,
0.41111293070407604356287744136, 1.16335135005503495850480861604, 3.09623664988971903387697038575, 3.60957613195605337441781942262, 4.28393317828554039360291719293, 5.10130315669302792826060765507, 5.76441502363501353611693013313, 6.52219115723883481619637247478, 7.12855129968799656165261561335, 7.40657053790604045136986154470
