| L(s) = 1 | − 0.193i·2-s + 1.96·4-s + (1.48 − 1.67i)5-s − i·7-s − 0.768i·8-s + (−0.324 − 0.287i)10-s − 2·11-s − 1.35i·13-s − 0.193·14-s + 3.77·16-s + 3.35i·17-s − 5.35·19-s + (2.90 − 3.28i)20-s + 0.387i·22-s + 4.96i·23-s + ⋯ |
| L(s) = 1 | − 0.137i·2-s + 0.981·4-s + (0.662 − 0.749i)5-s − 0.377i·7-s − 0.271i·8-s + (−0.102 − 0.0908i)10-s − 0.603·11-s − 0.374i·13-s − 0.0518·14-s + 0.943·16-s + 0.812i·17-s − 1.22·19-s + (0.649 − 0.735i)20-s + 0.0826i·22-s + 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59903 - 0.605565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59903 - 0.605565i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.48 + 1.67i)T \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 + 0.193iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 + 0.775iT - 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 9.92iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−11.54711143403294346003863887302, −10.45152688017041292222651583827, −10.04245753627899168736783865857, −8.619312045962259727983070150155, −7.80473498180319264562137744432, −6.54154200088920566720403510904, −5.74527851187100846863655285915, −4.44726406318928226945328405331, −2.83943926559673980971448900499, −1.48489269016061561210805772777,
2.14036571969006536939758190979, 2.97186615378028711966119657503, 4.88246346607067746871300579201, 6.20119128337941293824498121931, 6.68384503766088323788447013306, 7.81695234526073576556459547553, 8.927176020393277035626427789616, 10.25050808204526197270382750190, 10.66563104304813998810307056329, 11.72407332563675435400719163852
