| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 13-s + 15-s − 6·17-s + 4·19-s − 4·21-s + 4·23-s + 25-s − 27-s − 6·29-s − 8·31-s − 4·35-s + 2·37-s + 39-s − 6·41-s + 12·43-s − 45-s + 9·49-s + 6·51-s + 2·53-s − 4·57-s + 4·59-s + 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 9/7·49-s + 0.840·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.28005142473504, −12.90038434987858, −12.28981472056230, −11.84404031421077, −11.35712984408925, −11.04365247090834, −10.86317226890639, −10.18695678886096, −9.484608480297079, −8.940777078682213, −8.771552778964966, −7.880085576554237, −7.657746293448343, −7.085502002288300, −6.806508225177318, −5.847806828418802, −5.547629014588325, −4.925515045946995, −4.630911593087855, −4.032242440823194, −3.526846696089432, −2.643830376773300, −2.049830365997050, −1.500093497443073, −0.8010383867010077, 0,
0.8010383867010077, 1.500093497443073, 2.049830365997050, 2.643830376773300, 3.526846696089432, 4.032242440823194, 4.630911593087855, 4.925515045946995, 5.547629014588325, 5.847806828418802, 6.806508225177318, 7.085502002288300, 7.657746293448343, 7.880085576554237, 8.771552778964966, 8.940777078682213, 9.484608480297079, 10.18695678886096, 10.86317226890639, 11.04365247090834, 11.35712984408925, 11.84404031421077, 12.28981472056230, 12.90038434987858, 13.28005142473504
