| L(s) = 1 | + 3-s − 4·5-s − 2·7-s + 9-s + 6·13-s − 4·15-s + 17-s + 4·19-s − 2·21-s − 6·23-s + 11·25-s + 27-s + 4·29-s + 6·31-s + 8·35-s − 4·37-s + 6·39-s + 10·41-s − 4·43-s − 4·45-s − 4·47-s − 3·49-s + 51-s − 2·53-s + 4·57-s − 12·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 1.66·13-s − 1.03·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 11/5·25-s + 0.192·27-s + 0.742·29-s + 1.07·31-s + 1.35·35-s − 0.657·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.140·51-s − 0.274·53-s + 0.529·57-s − 1.56·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.03809288831977, −13.59362827629289, −12.99924987027682, −12.47059651771800, −12.09170433838300, −11.63127489712683, −11.07300848871911, −10.72526145747921, −9.960079976139262, −9.596205282817385, −8.900796340880643, −8.388334581817235, −8.012092909630713, −7.751913148668748, −6.962387094425090, −6.518772267215581, −6.046143880651219, −5.210453698802950, −4.509907908640712, −3.955837399977172, −3.588324257042554, −3.156842818361416, −2.577727093436737, −1.431284686172814, −0.8629180104758154, 0,
0.8629180104758154, 1.431284686172814, 2.577727093436737, 3.156842818361416, 3.588324257042554, 3.955837399977172, 4.509907908640712, 5.210453698802950, 6.046143880651219, 6.518772267215581, 6.962387094425090, 7.751913148668748, 8.012092909630713, 8.388334581817235, 8.900796340880643, 9.596205282817385, 9.960079976139262, 10.72526145747921, 11.07300848871911, 11.63127489712683, 12.09170433838300, 12.47059651771800, 12.99924987027682, 13.59362827629289, 14.03809288831977
