L(s) = 1 | + (1.41 + 1.41i)7-s − i·9-s + (1.41 − 1.41i)23-s − 2·41-s + (1.41 + 1.41i)47-s + 3.00i·49-s + (1.41 − 1.41i)63-s − 81-s − 2i·89-s + (−1.41 + 1.41i)103-s + ⋯ |
L(s) = 1 | + (1.41 + 1.41i)7-s − i·9-s + (1.41 − 1.41i)23-s − 2·41-s + (1.41 + 1.41i)47-s + 3.00i·49-s + (1.41 − 1.41i)63-s − 81-s − 2i·89-s + (−1.41 + 1.41i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.517383382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517383382\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 2T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.781184462414268596988684930658, −8.386609224501681794404328553973, −7.42634106389067589965179415152, −6.54937887168368387607143260012, −5.81497657749248190330483104458, −5.02880261823992864665981694581, −4.41164773945431332497170434520, −3.15620969394984185239991623342, −2.34654076833839008292226682875, −1.25049665581412941974710719706,
1.19846593088660010545087102603, 2.02394635151992678945856159291, 3.37693534274857280233526153234, 4.24170447864360398614841771417, 5.03045363450171316178310636004, 5.45490015785996835775925748502, 7.00618205700576076060867588322, 7.23206686480450800070882513957, 8.079097557460132601720875935582, 8.558521724688104867953923222011