L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 2.12·11-s − 12-s − 6.68·13-s − 14-s + 15-s + 16-s − 5·17-s + 18-s − 3.56·19-s − 20-s + 21-s + 2.12·22-s − 2.43·23-s − 24-s + 25-s − 6.68·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.640·11-s − 0.288·12-s − 1.85·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s − 0.817·19-s − 0.223·20-s + 0.218·21-s + 0.452·22-s − 0.508·23-s − 0.204·24-s + 0.200·25-s − 1.31·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 6.68T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 0.561T + 31T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 + 0.315T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 6.68T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.652420714269278564686753285487, −8.553488358181504791101811526665, −7.50335305222625085181300380662, −6.70137084399848694794174088545, −6.16905853612040079740785796708, −4.76951591052672606605992861028, −4.50612303761985982714189281352, −3.19765339415750741418118978281, −2.01926901614798684409134347927, 0,
2.01926901614798684409134347927, 3.19765339415750741418118978281, 4.50612303761985982714189281352, 4.76951591052672606605992861028, 6.16905853612040079740785796708, 6.70137084399848694794174088545, 7.50335305222625085181300380662, 8.553488358181504791101811526665, 9.652420714269278564686753285487