L(s) = 1 | − 2-s − 2.56·3-s + 4-s + 5-s + 2.56·6-s − 3·7-s − 8-s + 3.56·9-s − 10-s − 11-s − 2.56·12-s + 3.56·13-s + 3·14-s − 2.56·15-s + 16-s + 1.43·17-s − 3.56·18-s − 19-s + 20-s + 7.68·21-s + 22-s − 9.12·23-s + 2.56·24-s + 25-s − 3.56·26-s − 1.43·27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.447·5-s + 1.04·6-s − 1.13·7-s − 0.353·8-s + 1.18·9-s − 0.316·10-s − 0.301·11-s − 0.739·12-s + 0.987·13-s + 0.801·14-s − 0.661·15-s + 0.250·16-s + 0.348·17-s − 0.839·18-s − 0.229·19-s + 0.223·20-s + 1.67·21-s + 0.213·22-s − 1.90·23-s + 0.522·24-s + 0.200·25-s − 0.698·26-s − 0.276·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 9.12T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 + 0.438T + 41T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 3.56T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.007580079804110809689623253868, −6.86369039113355235338629589299, −6.50607968174673415436461760303, −5.86453928052456982023723479895, −5.39471985859703678539391523503, −4.21426822196787694833387758925, −3.30863542701167587921739800369, −2.15012166554226177636428758754, −0.986694571237062052272526580458, 0,
0.986694571237062052272526580458, 2.15012166554226177636428758754, 3.30863542701167587921739800369, 4.21426822196787694833387758925, 5.39471985859703678539391523503, 5.86453928052456982023723479895, 6.50607968174673415436461760303, 6.86369039113355235338629589299, 8.007580079804110809689623253868