L(s) = 1 | − 1.73i·2-s + 2·3-s − 0.999·4-s + 1.73i·5-s − 3.46i·6-s − 1.73i·8-s + 9-s + 2.99·10-s − 1.99·12-s + 3.46i·15-s − 5·16-s − 3·17-s − 1.73i·18-s + 3.46i·19-s − 1.73i·20-s + ⋯ |
L(s) = 1 | − 1.22i·2-s + 1.15·3-s − 0.499·4-s + 0.774i·5-s − 1.41i·6-s − 0.612i·8-s + 0.333·9-s + 0.948·10-s − 0.577·12-s + 0.894i·15-s − 1.25·16-s − 0.727·17-s − 0.408i·18-s + 0.794i·19-s − 0.387i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26684 - 0.952866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26684 - 0.952866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 8.66iT - 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−12.48263861716559487451818949027, −11.52441054296548637032148096270, −10.56403233314047195907411460948, −9.780212139297250293402784529936, −8.735107569485189719239757586320, −7.58992859012226347336486584256, −6.30914840314477631398284782079, −4.11034874790725181817896976209, −3.06189891578492420096607173424, −2.10572921794485249831794009614,
2.44501112123847370636860148022, 4.30239335282076746516779963742, 5.57603028151523584184682147069, 6.86820121374391796497923239343, 7.939310022246744940027896702216, 8.673371113803740723619786588554, 9.332238110505233578204456023620, 10.94847564488489828881242774172, 12.26818101199900444750312981580, 13.44476425929559468346538491569