L(s) = 1 | − 2-s + 3-s + 4-s − 0.453·5-s − 6-s + 3.87·7-s − 8-s + 9-s + 0.453·10-s − 1.29·11-s + 12-s − 13-s − 3.87·14-s − 0.453·15-s + 16-s − 0.204·17-s − 18-s − 0.475·19-s − 0.453·20-s + 3.87·21-s + 1.29·22-s + 1.84·23-s − 24-s − 4.79·25-s + 26-s + 27-s + 3.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.202·5-s − 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s + 0.143·10-s − 0.389·11-s + 0.288·12-s − 0.277·13-s − 1.03·14-s − 0.117·15-s + 0.250·16-s − 0.0494·17-s − 0.235·18-s − 0.109·19-s − 0.101·20-s + 0.846·21-s + 0.275·22-s + 0.385·23-s − 0.204·24-s − 0.958·25-s + 0.196·26-s + 0.192·27-s + 0.732·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.453T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 17 | \( 1 + 0.204T + 17T^{2} \) |
| 19 | \( 1 + 0.475T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 + 7.72T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 0.678T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 0.0379T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 2.30T + 97T^{2} \) |
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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
−7.79671118859045016015539065569, −7.17957356153019319365577347667, −6.19241277065225228640998668046, −5.33605452150117573633874707473, −4.68158310312501095116618109390, −3.84456083495646570633400579273, −2.93390962901739565942721000094, −1.96983854029079652288219520623, −1.48464769958654406166094910247, 0,
1.48464769958654406166094910247, 1.96983854029079652288219520623, 2.93390962901739565942721000094, 3.84456083495646570633400579273, 4.68158310312501095116618109390, 5.33605452150117573633874707473, 6.19241277065225228640998668046, 7.17957356153019319365577347667, 7.79671118859045016015539065569