| L(s) = 1 | − i·3-s + 4i·5-s − 7-s − 9-s − 2i·11-s + 6i·13-s + 4·15-s − 4·17-s − 4i·19-s + i·21-s + 2·23-s − 11·25-s + i·27-s + 2i·29-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.78i·5-s − 0.377·7-s − 0.333·9-s − 0.603i·11-s + 1.66i·13-s + 1.03·15-s − 0.970·17-s − 0.917i·19-s + 0.218i·21-s + 0.417·23-s − 2.20·25-s + 0.192i·27-s + 0.371i·29-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 4iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.70834149207927676619920451311, −6.85782672950787220939322583308, −6.67616073900524526108472432769, −6.21294747421646970658788295130, −5.02322372623098884776868496175, −4.02405480876161718604589134660, −3.17807122904881126005244579890, −2.55507776657246030308401610618, −1.71958472548994670981627071126, 0,
1.05352039163140314541465274874, 2.19219352449159311691295553007, 3.34378273326167599288314235215, 4.16101884963906831827577296676, 4.81710203151857549674279171297, 5.43515134825974356666601140719, 5.99617407797385932841893441289, 7.09611571090695418765319034525, 8.116851159532703854246143445539
