| L(s) = 1 | − 3-s + 5-s + 9-s − 2·13-s − 15-s + 6·17-s − 8·23-s + 25-s − 27-s + 2·29-s + 4·31-s − 10·37-s + 2·39-s − 2·41-s + 4·43-s + 45-s − 8·47-s − 7·49-s − 6·51-s + 2·53-s + 8·59-s − 2·61-s − 2·65-s − 12·67-s + 8·69-s + 8·71-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s + 0.274·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s − 1.46·67-s + 0.963·69-s + 0.949·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.40376485565395, −12.91724859242027, −12.36870261671237, −11.96624891554497, −11.75756393934463, −11.12380213947435, −10.29176497440690, −10.18047728199586, −9.919580777981795, −9.219606649669765, −8.642437349824966, −8.081466424101773, −7.655927482666633, −7.128435717773697, −6.525835879554505, −6.083834489429642, −5.598885733781869, −5.143775771441787, −4.614237265532683, −4.021520311150569, −3.324061977211706, −2.879287712550525, −1.961905951512815, −1.614483425881142, −0.7730520285231310, 0,
0.7730520285231310, 1.614483425881142, 1.961905951512815, 2.879287712550525, 3.324061977211706, 4.021520311150569, 4.614237265532683, 5.143775771441787, 5.598885733781869, 6.083834489429642, 6.525835879554505, 7.128435717773697, 7.655927482666633, 8.081466424101773, 8.642437349824966, 9.219606649669765, 9.919580777981795, 10.18047728199586, 10.29176497440690, 11.12380213947435, 11.75756393934463, 11.96624891554497, 12.36870261671237, 12.91724859242027, 13.40376485565395
