| L(s) = 1 | − 3-s + 9-s + 13-s + 6·17-s − 4·23-s − 27-s − 10·29-s + 6·37-s − 39-s + 2·41-s − 4·43-s − 7·49-s − 6·51-s + 6·53-s + 6·61-s + 4·67-s + 4·69-s − 16·71-s + 2·73-s + 81-s + 4·83-s + 10·87-s − 6·89-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.277·13-s + 1.45·17-s − 0.834·23-s − 0.192·27-s − 1.85·29-s + 0.986·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 49-s − 0.840·51-s + 0.824·53-s + 0.768·61-s + 0.488·67-s + 0.481·69-s − 1.89·71-s + 0.234·73-s + 1/9·81-s + 0.439·83-s + 1.07·87-s − 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.47548723830503, −15.84485196981416, −15.13983938963473, −14.61024251860321, −14.18745088840695, −13.31147981138233, −12.98137366253306, −12.33138182419763, −11.67794706537752, −11.39317860801128, −10.60902558153747, −10.07723362605720, −9.563986910424351, −8.948003033356973, −7.977508056779610, −7.751453674617639, −6.940266277295014, −6.252244408197993, −5.598268782116823, −5.266575094330092, −4.228920677506390, −3.738123448034166, −2.901537074707757, −1.884384203671139, −1.109858494506879, 0,
1.109858494506879, 1.884384203671139, 2.901537074707757, 3.738123448034166, 4.228920677506390, 5.266575094330092, 5.598268782116823, 6.252244408197993, 6.940266277295014, 7.751453674617639, 7.977508056779610, 8.948003033356973, 9.563986910424351, 10.07723362605720, 10.60902558153747, 11.39317860801128, 11.67794706537752, 12.33138182419763, 12.98137366253306, 13.31147981138233, 14.18745088840695, 14.61024251860321, 15.13983938963473, 15.84485196981416, 16.47548723830503
