| L(s) = 1 | + 5-s + 4·7-s − 11-s + 2·13-s + 17-s − 2·19-s + 6·23-s + 25-s + 6·29-s + 4·31-s + 4·35-s − 10·37-s + 4·43-s − 12·47-s + 9·49-s − 55-s − 10·61-s + 2·65-s − 14·67-s − 12·71-s + 2·73-s − 4·77-s − 2·79-s + 85-s − 6·89-s + 8·91-s − 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.301·11-s + 0.554·13-s + 0.242·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.134·55-s − 1.28·61-s + 0.248·65-s − 1.71·67-s − 1.42·71-s + 0.234·73-s − 0.455·77-s − 0.225·79-s + 0.108·85-s − 0.635·89-s + 0.838·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.64235938876273, −13.37634311486984, −12.76472163706806, −12.06186720926499, −11.91264610478619, −11.17529929910741, −10.71097372151702, −10.57070476680682, −9.876922353713628, −9.249349845344957, −8.715943565916723, −8.308507479484863, −8.011574269151090, −7.216383976851098, −6.884556613460373, −6.166862851847129, −5.677842711070931, −5.087459260821175, −4.647253552595631, −4.305193674359151, −3.296877338604122, −2.904640770378991, −2.137149658355778, −1.414656823194990, −1.190047547924240, 0,
1.190047547924240, 1.414656823194990, 2.137149658355778, 2.904640770378991, 3.296877338604122, 4.305193674359151, 4.647253552595631, 5.087459260821175, 5.677842711070931, 6.166862851847129, 6.884556613460373, 7.216383976851098, 8.011574269151090, 8.308507479484863, 8.715943565916723, 9.249349845344957, 9.876922353713628, 10.57070476680682, 10.71097372151702, 11.17529929910741, 11.91264610478619, 12.06186720926499, 12.76472163706806, 13.37634311486984, 13.64235938876273
