L(s) = 1 | + 2-s + (−1 − i)3-s + 4-s + (−2 − i)5-s + (−1 − i)6-s + (3 − 3i)7-s + 8-s − i·9-s + (−2 − i)10-s + (1 + i)11-s + (−1 − i)12-s + 4i·13-s + (3 − 3i)14-s + (1 + 3i)15-s + 16-s + (−1 − 4i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.577 − 0.577i)3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + (−0.408 − 0.408i)6-s + (1.13 − 1.13i)7-s + 0.353·8-s − 0.333i·9-s + (−0.632 − 0.316i)10-s + (0.301 + 0.301i)11-s + (−0.288 − 0.288i)12-s + 1.10i·13-s + (0.801 − 0.801i)14-s + (0.258 + 0.774i)15-s + 0.250·16-s + (−0.242 − 0.970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16423 - 0.735243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16423 - 0.735243i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + i)T \) |
| 17 | \( 1 + (1 + 4i)T \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 + (-7 + 7i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1 + i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + (9 + 9i)T + 61iT^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + (-1 + i)T - 71iT^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 + (-3 - 3i)T + 79iT^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (5 + 5i)T + 97iT^{2} \) |
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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
−12.30872379070584069272965824626, −11.74212759132257079134006454488, −11.21623606705598553803781852991, −9.706844422451958685104898079194, −8.016811289869478062046925192278, −7.33478917357118335549553531501, −6.21918428108980912645740534679, −4.65670264009737724695989235879, −3.96435464936665340622172962678, −1.36130075350983249833438177699,
2.65378181486934291426092448968, 4.31746307178271711054434627085, 5.17416870779593159403904206997, 6.28219970826725314459548453440, 7.84169226384929036796104729059, 8.637831267226108235874119331219, 10.58879304575573474855103256916, 10.98984956864593493188549839051, 11.90744628447711914546163159567, 12.66834478168736826778636778428