L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + i·7-s − i·8-s − 9-s − 4·11-s + i·12-s − 2i·13-s − 14-s + 16-s + 6i·17-s − i·18-s + 21-s − 4i·22-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.218·21-s − 0.852i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.323845620263769474202194793591, −8.476667599480528572984716795808, −7.87554170069646117703827877949, −7.15479985705976186082730493055, −6.01192522266309615903561756089, −5.60609386748082025149710631782, −4.44883157320725270432133685311, −3.16112833952143115810703523149, −1.91466993522813114027894941489, 0,
1.88864307716317065101849509719, 3.10998675331236431894781811691, 3.92928183227287822437359444945, 5.07252470279251523848864609082, 5.56965946779049544365706938216, 7.17199259741541919247383076839, 7.75119734905749421249091080610, 8.998605946800609538118590352237, 9.525039483962604332555282034981