| L(s) = 1 | − 0.471·3-s − 3.53·5-s − 2.29·7-s − 2.77·9-s + 0.762·13-s + 1.66·15-s + 1.00·17-s + 19-s + 1.08·21-s − 2.94·23-s + 7.50·25-s + 2.72·27-s − 5.04·29-s + 3.30·31-s + 8.13·35-s + 7.77·37-s − 0.359·39-s − 1.35·41-s + 3.91·43-s + 9.82·45-s + 0.474·47-s − 1.71·49-s − 0.475·51-s − 5.38·53-s − 0.471·57-s − 1.88·59-s − 2.99·61-s + ⋯ |
| L(s) = 1 | − 0.272·3-s − 1.58·5-s − 0.869·7-s − 0.925·9-s + 0.211·13-s + 0.430·15-s + 0.244·17-s + 0.229·19-s + 0.236·21-s − 0.615·23-s + 1.50·25-s + 0.523·27-s − 0.936·29-s + 0.593·31-s + 1.37·35-s + 1.27·37-s − 0.0575·39-s − 0.211·41-s + 0.596·43-s + 1.46·45-s + 0.0692·47-s − 0.244·49-s − 0.0665·51-s − 0.739·53-s − 0.0624·57-s − 0.245·59-s − 0.383·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.471T + 3T^{2} \) |
| 5 | \( 1 + 3.53T + 5T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 13 | \( 1 - 0.762T + 13T^{2} \) |
| 17 | \( 1 - 1.00T + 17T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 + 5.04T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 7.77T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 - 0.474T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 5.84T + 71T^{2} \) |
| 73 | \( 1 - 8.06T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−7.58116542325710279180978458052, −6.60434432127830608518732550613, −6.14850521856966402160818302586, −5.31104871949185299205852317955, −4.50514109002281534340465209213, −3.68904913141294262655327222184, −3.28878512613930065986847171650, −2.38937127103862155732867261664, −0.827231966236917548893811304463, 0,
0.827231966236917548893811304463, 2.38937127103862155732867261664, 3.28878512613930065986847171650, 3.68904913141294262655327222184, 4.50514109002281534340465209213, 5.31104871949185299205852317955, 6.14850521856966402160818302586, 6.60434432127830608518732550613, 7.58116542325710279180978458052
