L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.34 − 1.22i)7-s + 0.999i·8-s + (2.03 + 1.17i)11-s + 4.64i·13-s + (2.64 − 0.107i)14-s + (−0.5 − 0.866i)16-s + (2.28 − 3.95i)17-s + (0.491 − 0.283i)19-s − 2.35·22-s + (−5.04 + 2.91i)23-s + (−2.32 − 4.02i)26-s + (−2.23 + 1.41i)28-s − 2.55i·29-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.885 − 0.464i)7-s + 0.353i·8-s + (0.615 + 0.355i)11-s + 1.28i·13-s + (0.706 − 0.0287i)14-s + (−0.125 − 0.216i)16-s + (0.553 − 0.958i)17-s + (0.112 − 0.0650i)19-s − 0.502·22-s + (−1.05 + 0.607i)23-s + (−0.455 − 0.789i)26-s + (−0.422 + 0.267i)28-s − 0.474i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6188054093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6188054093\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.34 + 1.22i)T \) |
good | 11 | \( 1 + (-2.03 - 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.64iT - 13T^{2} \) |
| 17 | \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.491 + 0.283i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.04 - 2.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.55iT - 29T^{2} \) |
| 31 | \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.63 - 8.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.68T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + (3.15 + 5.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.5 + 6.07i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.67 + 2.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.02iT - 71T^{2} \) |
| 73 | \( 1 + (-7.11 - 4.10i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.13 - 7.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.171T + 83T^{2} \) |
| 89 | \( 1 + (2.72 + 4.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−9.237873783047053085251389950547, −8.094083131529514068828651128969, −7.49672472472026640661873033551, −6.60964809636196808771456174489, −6.37806603756571573228344138942, −5.22365241811826377764322361121, −4.27408693513286024886777837946, −3.46245305067011653677358439100, −2.27240291039242667920406478882, −1.15242760817457325700261456645,
0.25924861545441536925406195797, 1.53370053170960998076285950904, 2.75486832719398140029264912632, 3.40258196109789578269905198775, 4.26407550317351241295857695648, 5.69877711594075047317201285206, 6.03687616750445101992757701111, 6.95703814533600182007046882755, 7.961380576660756473583981280127, 8.323967857511365171762690629212