L(s) = 1 | − 2.89·3-s + 5-s + 0.451·7-s + 5.36·9-s − 4.25·11-s − 4.52·13-s − 2.89·15-s + 2.97·17-s − 0.416·19-s − 1.30·21-s + 4.36·23-s + 25-s − 6.84·27-s − 1.71·29-s + 0.519·31-s + 12.3·33-s + 0.451·35-s + 5.90·37-s + 13.0·39-s + 7.30·41-s + 6.72·43-s + 5.36·45-s − 7.34·47-s − 6.79·49-s − 8.60·51-s − 8.53·53-s − 4.25·55-s + ⋯ |
L(s) = 1 | − 1.66·3-s + 0.447·5-s + 0.170·7-s + 1.78·9-s − 1.28·11-s − 1.25·13-s − 0.746·15-s + 0.721·17-s − 0.0955·19-s − 0.284·21-s + 0.910·23-s + 0.200·25-s − 1.31·27-s − 0.318·29-s + 0.0932·31-s + 2.14·33-s + 0.0762·35-s + 0.971·37-s + 2.09·39-s + 1.14·41-s + 1.02·43-s + 0.799·45-s − 1.07·47-s − 0.970·49-s − 1.20·51-s − 1.17·53-s − 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 - 0.451T + 7T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 19 | \( 1 + 0.416T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 - 0.519T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 7.30T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 8.53T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 - 9.96T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.52990465899953410354244284908, −6.97827500417358057047430758977, −6.06027581866459434158160302243, −5.62203125917150596288131320366, −4.87051896255424216260017970092, −4.60099048680208281570902220352, −3.13149173451312533693668381102, −2.22673491102142053037020735590, −1.03833463078192215057342192094, 0,
1.03833463078192215057342192094, 2.22673491102142053037020735590, 3.13149173451312533693668381102, 4.60099048680208281570902220352, 4.87051896255424216260017970092, 5.62203125917150596288131320366, 6.06027581866459434158160302243, 6.97827500417358057047430758977, 7.52990465899953410354244284908