L(s) = 1 | − 2.20·3-s − 3.50·5-s − 0.937·7-s + 1.88·9-s − 1.06·13-s + 7.74·15-s + 0.235·17-s − 0.882·19-s + 2.07·21-s − 0.827·23-s + 7.28·25-s + 2.46·27-s − 2.13·29-s − 2.07·31-s + 3.28·35-s − 8.23·37-s + 2.34·39-s + 10.0·41-s + 6.86·43-s − 6.59·45-s + 3.83·47-s − 6.12·49-s − 0.520·51-s + 7.19·53-s + 1.94·57-s + 7.31·59-s + 7.30·61-s + ⋯ |
L(s) = 1 | − 1.27·3-s − 1.56·5-s − 0.354·7-s + 0.627·9-s − 0.294·13-s + 1.99·15-s + 0.0571·17-s − 0.202·19-s + 0.452·21-s − 0.172·23-s + 1.45·25-s + 0.475·27-s − 0.397·29-s − 0.372·31-s + 0.555·35-s − 1.35·37-s + 0.375·39-s + 1.57·41-s + 1.04·43-s − 0.983·45-s + 0.559·47-s − 0.874·49-s − 0.0728·51-s + 0.987·53-s + 0.258·57-s + 0.952·59-s + 0.935·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 71 | \( 1 - T \) |
good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 0.937T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 0.235T + 17T^{2} \) |
| 19 | \( 1 + 0.882T + 19T^{2} \) |
| 23 | \( 1 + 0.827T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 + 2.07T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 - 7.30T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.86648652232019532599208101855, −7.11561759060002878508057875866, −6.63595105084633948152624586498, −5.66208551089175413739148154668, −5.09875697622815575950614267784, −4.15147357207714825157202829377, −3.67098558672581137395508576007, −2.49150565078587311090686401150, −0.864350649774805650297541042348, 0,
0.864350649774805650297541042348, 2.49150565078587311090686401150, 3.67098558672581137395508576007, 4.15147357207714825157202829377, 5.09875697622815575950614267784, 5.66208551089175413739148154668, 6.63595105084633948152624586498, 7.11561759060002878508057875866, 7.86648652232019532599208101855