L(s) = 1 | + 0.820·3-s + 0.0215·5-s + 7-s − 2.32·9-s − 11-s − 13-s + 0.0177·15-s + 0.645·17-s + 2.66·19-s + 0.820·21-s + 0.816·23-s − 4.99·25-s − 4.37·27-s − 2.98·29-s − 2.72·31-s − 0.820·33-s + 0.0215·35-s − 3.34·37-s − 0.820·39-s + 3.93·41-s − 4.28·43-s − 0.0501·45-s − 7.90·47-s + 49-s + 0.529·51-s + 7.61·53-s − 0.0215·55-s + ⋯ |
L(s) = 1 | + 0.473·3-s + 0.00964·5-s + 0.377·7-s − 0.775·9-s − 0.301·11-s − 0.277·13-s + 0.00457·15-s + 0.156·17-s + 0.610·19-s + 0.179·21-s + 0.170·23-s − 0.999·25-s − 0.841·27-s − 0.553·29-s − 0.488·31-s − 0.142·33-s + 0.00364·35-s − 0.549·37-s − 0.131·39-s + 0.614·41-s − 0.653·43-s − 0.00748·45-s − 1.15·47-s + 0.142·49-s + 0.0741·51-s + 1.04·53-s − 0.00290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.820T + 3T^{2} \) |
| 5 | \( 1 - 0.0215T + 5T^{2} \) |
| 17 | \( 1 - 0.645T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 - 0.816T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 7.90T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 8.01T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 0.993T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 4.20T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.012006658853959727746703223238, −7.58490103541845862823898774133, −6.69002131909679366477892525860, −5.65334807589791689659758751310, −5.26202231765568841247846891488, −4.16740084367364015199764531273, −3.32945874645382723163322438158, −2.53092810104642267733891672857, −1.57601418903372879387098370448, 0,
1.57601418903372879387098370448, 2.53092810104642267733891672857, 3.32945874645382723163322438158, 4.16740084367364015199764531273, 5.26202231765568841247846891488, 5.65334807589791689659758751310, 6.69002131909679366477892525860, 7.58490103541845862823898774133, 8.012006658853959727746703223238