| L(s) = 1 | − 5-s − 3·7-s + 3·11-s + 13-s + 3·17-s − 6·19-s + 4·23-s + 25-s − 29-s − 4·31-s + 3·35-s − 4·37-s − 2·41-s + 4·43-s + 3·47-s + 2·49-s − 6·53-s − 3·55-s + 10·59-s − 4·61-s − 65-s − 9·67-s + 12·71-s + 4·73-s − 9·77-s + 14·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.185·29-s − 0.718·31-s + 0.507·35-s − 0.657·37-s − 0.312·41-s + 0.609·43-s + 0.437·47-s + 2/7·49-s − 0.824·53-s − 0.404·55-s + 1.30·59-s − 0.512·61-s − 0.124·65-s − 1.09·67-s + 1.42·71-s + 0.468·73-s − 1.02·77-s + 1.57·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.82999347089749032002524118176, −6.97581278867617080176089594381, −6.52240527058937538600417208113, −5.81584015082331568938780501460, −4.87374882335068754123848938012, −3.85826409321470029780970608574, −3.51042092525369676638429564418, −2.47782508713054971289568363510, −1.25056308331003374559472083993, 0,
1.25056308331003374559472083993, 2.47782508713054971289568363510, 3.51042092525369676638429564418, 3.85826409321470029780970608574, 4.87374882335068754123848938012, 5.81584015082331568938780501460, 6.52240527058937538600417208113, 6.97581278867617080176089594381, 7.82999347089749032002524118176
