| L(s) = 1 | − 5-s − 2·7-s + 4·11-s − 2·13-s − 5·17-s + 5·19-s + 23-s + 25-s + 2·29-s − 7·31-s + 2·35-s − 6·37-s − 4·43-s + 4·47-s − 3·49-s − 9·53-s − 4·55-s + 14·59-s − 11·61-s + 2·65-s − 14·67-s − 12·73-s − 8·77-s + 3·79-s − 83-s + 5·85-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s − 1.21·17-s + 1.14·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.25·31-s + 0.338·35-s − 0.986·37-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 1.23·53-s − 0.539·55-s + 1.82·59-s − 1.40·61-s + 0.248·65-s − 1.71·67-s − 1.40·73-s − 0.911·77-s + 0.337·79-s − 0.109·83-s + 0.542·85-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.987750842725442006551373585122, −7.83561238793477898112972070751, −6.99079300610682494080704471036, −6.56644807720865578285045226703, −5.50896552798411729795788051275, −4.54531903488051111371354188613, −3.70443672612966276007439183421, −2.89711829476831056635952255938, −1.54336384157991496196857038231, 0,
1.54336384157991496196857038231, 2.89711829476831056635952255938, 3.70443672612966276007439183421, 4.54531903488051111371354188613, 5.50896552798411729795788051275, 6.56644807720865578285045226703, 6.99079300610682494080704471036, 7.83561238793477898112972070751, 8.987750842725442006551373585122
