L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.280 − 0.550i)7-s + (0.453 + 0.891i)8-s + (0.587 − 0.809i)9-s + (−0.309 − 0.951i)11-s + (−1.59 + 0.253i)13-s + (−0.587 − 0.190i)14-s + (0.809 − 0.587i)16-s + (−0.891 − 0.453i)18-s + (0.363 − 1.11i)19-s + (−0.891 + 0.453i)22-s + (1.34 + 1.34i)23-s + (0.5 + 1.53i)26-s + (−0.0966 + 0.610i)28-s + ⋯ |
L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.280 − 0.550i)7-s + (0.453 + 0.891i)8-s + (0.587 − 0.809i)9-s + (−0.309 − 0.951i)11-s + (−1.59 + 0.253i)13-s + (−0.587 − 0.190i)14-s + (0.809 − 0.587i)16-s + (−0.891 − 0.453i)18-s + (0.363 − 1.11i)19-s + (−0.891 + 0.453i)22-s + (1.34 + 1.34i)23-s + (0.5 + 1.53i)26-s + (−0.0966 + 0.610i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8482507196\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8482507196\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.280 + 0.550i)T + (-0.587 - 0.809i)T^{2} \) |
| 13 | \( 1 + (1.59 - 0.253i)T + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.04 + 0.533i)T + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (0.297 + 1.87i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + 0.618iT - T^{2} \) |
| 97 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
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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
−9.107086910812647671604238023576, −8.382368266881990017264491417301, −7.27738158299331365177475597478, −6.94030541269639242326648208723, −5.28826429499540858236111256867, −4.92001582621178972678591343822, −3.72907749689211163322751180040, −3.12899976763315658714357363327, −1.91820464276747382902037136638, −0.64071865367951959266472722569,
1.67245070529470855951599069731, 2.82928759752096356185715723531, 4.36919519532036948315886646953, 4.96706615251193945854946458461, 5.43988465072142914512303959008, 6.72129539185299290684875306555, 7.24053576525273195095932420758, 7.935939125929866030672996912104, 8.576777280929351508051626083664, 9.559618247609312371001545465665