| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 6·13-s − 15-s + 2·17-s − 4·19-s − 21-s + 23-s + 25-s − 27-s − 2·29-s + 4·31-s + 35-s + 2·37-s + 6·39-s − 6·41-s + 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 10·53-s + 4·57-s − 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.529·57-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 18 T + p T^{2} \) |
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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
−16.10685167248538, −15.30344028841974, −14.83395140750408, −14.51402991259428, −13.78672160773227, −13.15579691610677, −12.67500778142817, −12.02689217961899, −11.73212649905285, −10.96866580134843, −10.31707196402976, −10.03371462549750, −9.330468151110410, −8.762688841714378, −7.852607004806665, −7.557388865319552, −6.632441895834333, −6.355652797525556, −5.322108100415080, −5.089417754797937, −4.404184231686096, −3.581081748912200, −2.550182691111284, −2.058955264728666, −1.042626689485317, 0,
1.042626689485317, 2.058955264728666, 2.550182691111284, 3.581081748912200, 4.404184231686096, 5.089417754797937, 5.322108100415080, 6.355652797525556, 6.632441895834333, 7.557388865319552, 7.852607004806665, 8.762688841714378, 9.330468151110410, 10.03371462549750, 10.31707196402976, 10.96866580134843, 11.73212649905285, 12.02689217961899, 12.67500778142817, 13.15579691610677, 13.78672160773227, 14.51402991259428, 14.83395140750408, 15.30344028841974, 16.10685167248538
