L(s) = 1 | + 1.01·3-s + 3.66·5-s + 1.84·7-s − 1.97·9-s + 2.84·11-s + 4.78·13-s + 3.70·15-s − 7.85·17-s + 1.81·19-s + 1.87·21-s − 5.67·23-s + 8.42·25-s − 5.03·27-s + 7.12·29-s − 2.60·31-s + 2.87·33-s + 6.77·35-s + 4.12·37-s + 4.84·39-s + 10.0·41-s + 4.94·43-s − 7.24·45-s + 10.9·47-s − 3.57·49-s − 7.94·51-s − 12.8·53-s + 10.4·55-s + ⋯ |
L(s) = 1 | + 0.584·3-s + 1.63·5-s + 0.699·7-s − 0.658·9-s + 0.857·11-s + 1.32·13-s + 0.957·15-s − 1.90·17-s + 0.415·19-s + 0.408·21-s − 1.18·23-s + 1.68·25-s − 0.969·27-s + 1.32·29-s − 0.467·31-s + 0.501·33-s + 1.14·35-s + 0.678·37-s + 0.775·39-s + 1.56·41-s + 0.754·43-s − 1.07·45-s + 1.59·47-s − 0.511·49-s − 1.11·51-s − 1.76·53-s + 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.698934758\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.698934758\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 2.60T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 + 7.49T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 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
−8.621736198300086123783541111632, −7.946947712746172013901914512831, −6.77859738717610717107806834661, −6.10073808898428717187134638030, −5.75967013874344342293134725164, −4.61754061065205424972508429569, −3.87715752399051114583790416267, −2.64875664690098555396958460560, −2.07621593190165400611714883016, −1.17374136650908496040003955490,
1.17374136650908496040003955490, 2.07621593190165400611714883016, 2.64875664690098555396958460560, 3.87715752399051114583790416267, 4.61754061065205424972508429569, 5.75967013874344342293134725164, 6.10073808898428717187134638030, 6.77859738717610717107806834661, 7.946947712746172013901914512831, 8.621736198300086123783541111632