| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s − 4·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 6·37-s + 2·39-s − 10·41-s + 4·43-s − 45-s + 12·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s − 12·59-s − 2·61-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.10495596215027, −13.61621940519941, −13.11710447022055, −12.44128127236872, −12.35837293453863, −11.64784175634347, −11.24299324867990, −10.61722447185268, −10.35493378967841, −9.617972263118206, −9.405174912962410, −8.409547508265703, −8.067551605684031, −7.516706233475963, −7.203671440983774, −6.398158017018184, −5.867151898210536, −5.419929980096095, −4.756902163769971, −4.406490404119228, −3.598784742985754, −3.024544810737023, −2.327397399890202, −1.653683914387751, −0.6654256230701555, 0,
0.6654256230701555, 1.653683914387751, 2.327397399890202, 3.024544810737023, 3.598784742985754, 4.406490404119228, 4.756902163769971, 5.419929980096095, 5.867151898210536, 6.398158017018184, 7.203671440983774, 7.516706233475963, 8.067551605684031, 8.409547508265703, 9.405174912962410, 9.617972263118206, 10.35493378967841, 10.61722447185268, 11.24299324867990, 11.64784175634347, 12.35837293453863, 12.44128127236872, 13.11710447022055, 13.61621940519941, 14.10495596215027
