L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3.08·7-s − 8-s + 9-s − 10-s + 1.80·11-s + 12-s − 5.95·13-s + 3.08·14-s + 15-s + 16-s − 18-s − 8.33·19-s + 20-s − 3.08·21-s − 1.80·22-s + 4.61·23-s − 24-s + 25-s + 5.95·26-s + 27-s − 3.08·28-s − 9.88·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.16·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.544·11-s + 0.288·12-s − 1.65·13-s + 0.825·14-s + 0.258·15-s + 0.250·16-s − 0.235·18-s − 1.91·19-s + 0.223·20-s − 0.673·21-s − 0.385·22-s + 0.963·23-s − 0.204·24-s + 0.200·25-s + 1.16·26-s + 0.192·27-s − 0.583·28-s − 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.061966789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.061966789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 19 | \( 1 + 8.33T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 + 8.72T + 31T^{2} \) |
| 37 | \( 1 + 8.97T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.765T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 + 0.639T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.64235713290801854893077811742, −7.13800449174003860416617656115, −6.66765441916147127067794570204, −5.86504196284601320788879423209, −5.08640483193377224015494907339, −3.99578914200578305972484755975, −3.39449748227576334653663819886, −2.28275802113924412836202952226, −2.07252012861145091632146763297, −0.50658212110035418782352350087,
0.50658212110035418782352350087, 2.07252012861145091632146763297, 2.28275802113924412836202952226, 3.39449748227576334653663819886, 3.99578914200578305972484755975, 5.08640483193377224015494907339, 5.86504196284601320788879423209, 6.66765441916147127067794570204, 7.13800449174003860416617656115, 7.64235713290801854893077811742