L(s) = 1 | − 2-s − 2.04·3-s + 4-s − 5-s + 2.04·6-s − 7-s − 8-s + 1.18·9-s + 10-s − 2.04·12-s − 3.50·13-s + 14-s + 2.04·15-s + 16-s + 4.28·17-s − 1.18·18-s − 4.44·19-s − 20-s + 2.04·21-s − 2·23-s + 2.04·24-s + 25-s + 3.50·26-s + 3.70·27-s − 28-s − 1.30·29-s − 2.04·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.18·3-s + 0.5·4-s − 0.447·5-s + 0.835·6-s − 0.377·7-s − 0.353·8-s + 0.396·9-s + 0.316·10-s − 0.590·12-s − 0.971·13-s + 0.267·14-s + 0.528·15-s + 0.250·16-s + 1.03·17-s − 0.280·18-s − 1.02·19-s − 0.223·20-s + 0.446·21-s − 0.417·23-s + 0.417·24-s + 0.200·25-s + 0.686·26-s + 0.713·27-s − 0.188·28-s − 0.242·29-s − 0.373·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 0.327T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 + 5.87T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 + 0.498T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−7.36402212856831491653661156963, −6.89386399912371742066855801629, −5.95646203867408596420462089869, −5.66373512038698636977936137286, −4.73545005867717265065072720047, −3.95122225716364527548652146819, −2.97376422352147085524106343460, −2.05301745098409672903058895852, −0.811222802022697145909076791419, 0,
0.811222802022697145909076791419, 2.05301745098409672903058895852, 2.97376422352147085524106343460, 3.95122225716364527548652146819, 4.73545005867717265065072720047, 5.66373512038698636977936137286, 5.95646203867408596420462089869, 6.89386399912371742066855801629, 7.36402212856831491653661156963