| L(s) = 1 | + 3·7-s + 4·11-s − 13-s + 4·17-s − 19-s − 4·23-s − 4·31-s + 9·37-s + 8·43-s + 12·47-s + 2·49-s + 8·53-s + 4·59-s − 5·61-s − 11·67-s + 8·71-s − 73-s + 12·77-s − 5·79-s − 8·83-s + 12·89-s − 3·91-s − 5·97-s − 103-s − 12·107-s − 14·109-s − 12·113-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s − 0.718·31-s + 1.47·37-s + 1.21·43-s + 1.75·47-s + 2/7·49-s + 1.09·53-s + 0.520·59-s − 0.640·61-s − 1.34·67-s + 0.949·71-s − 0.117·73-s + 1.36·77-s − 0.562·79-s − 0.878·83-s + 1.27·89-s − 0.314·91-s − 0.507·97-s − 0.0985·103-s − 1.16·107-s − 1.34·109-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.606064765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606064765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.008995197696269880378971545448, −7.62584756382192034595731095028, −6.80329678780076868697686975674, −5.89593616531456502959191451729, −5.38826294062667844392903271367, −4.28218480584089593637242131239, −3.99224601439891212375851541346, −2.73984235282526391923240573925, −1.78322332963611061965790145365, −0.931650633177113506766391107325,
0.931650633177113506766391107325, 1.78322332963611061965790145365, 2.73984235282526391923240573925, 3.99224601439891212375851541346, 4.28218480584089593637242131239, 5.38826294062667844392903271367, 5.89593616531456502959191451729, 6.80329678780076868697686975674, 7.62584756382192034595731095028, 8.008995197696269880378971545448
