L(s) = 1 | − 5-s + 7-s + 6·11-s − 2·13-s + 3·17-s − 6·19-s − 23-s + 25-s − 3·29-s − 3·31-s − 35-s + 37-s − 9·41-s − 8·43-s − 4·47-s − 6·49-s − 53-s − 6·55-s − 59-s + 8·61-s + 2·65-s − 7·67-s + 5·71-s − 6·73-s + 6·77-s + 11·83-s − 3·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.80·11-s − 0.554·13-s + 0.727·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s − 0.169·35-s + 0.164·37-s − 1.40·41-s − 1.21·43-s − 0.583·47-s − 6/7·49-s − 0.137·53-s − 0.809·55-s − 0.130·59-s + 1.02·61-s + 0.248·65-s − 0.855·67-s + 0.593·71-s − 0.702·73-s + 0.683·77-s + 1.20·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.44268901059253711967838281422, −6.71540578696026715311563108121, −6.28369989029524392041558534909, −5.28644711767548297022964285101, −4.57946370959352281616336665602, −3.86558075722039564519680808585, −3.30610986537926236435350633414, −2.04029743175111075742285475110, −1.35546198204889795262475369857, 0,
1.35546198204889795262475369857, 2.04029743175111075742285475110, 3.30610986537926236435350633414, 3.86558075722039564519680808585, 4.57946370959352281616336665602, 5.28644711767548297022964285101, 6.28369989029524392041558534909, 6.71540578696026715311563108121, 7.44268901059253711967838281422