| L(s) = 1 | − 0.959·3-s − 4.10·7-s − 2.07·9-s + 5.18·11-s + 0.586·13-s + 2.10·17-s + 0.372·19-s + 3.94·21-s − 6.13·23-s + 4.87·27-s + 29-s − 1.36·31-s − 4.97·33-s + 2.71·37-s − 0.563·39-s + 5.64·41-s − 11.0·43-s + 8.36·47-s + 9.88·49-s − 2.02·51-s − 3.16·53-s − 0.357·57-s − 6.58·59-s + 2.28·61-s + 8.54·63-s − 7.61·67-s + 5.88·69-s + ⋯ |
| L(s) = 1 | − 0.553·3-s − 1.55·7-s − 0.693·9-s + 1.56·11-s + 0.162·13-s + 0.511·17-s + 0.0854·19-s + 0.860·21-s − 1.27·23-s + 0.937·27-s + 0.185·29-s − 0.245·31-s − 0.866·33-s + 0.447·37-s − 0.0901·39-s + 0.881·41-s − 1.67·43-s + 1.21·47-s + 1.41·49-s − 0.283·51-s − 0.434·53-s − 0.0473·57-s − 0.857·59-s + 0.292·61-s + 1.07·63-s − 0.930·67-s + 0.708·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.959T + 3T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 13 | \( 1 - 0.586T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 0.372T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−7.67734944105620314132917854558, −6.78978732433391076891892221028, −6.16155651082660923619328629248, −6.00159623804258317158632936845, −4.91967737918241201298685426442, −3.84382035619640918972022036714, −3.43882881583942501993216668313, −2.42391218761539819787294484568, −1.10596454577136408594700761267, 0,
1.10596454577136408594700761267, 2.42391218761539819787294484568, 3.43882881583942501993216668313, 3.84382035619640918972022036714, 4.91967737918241201298685426442, 6.00159623804258317158632936845, 6.16155651082660923619328629248, 6.78978732433391076891892221028, 7.67734944105620314132917854558
