| L(s) = 1 | − 3.55i·2-s − 8.63·4-s + 0.118i·5-s − 12.0·7-s + 16.4i·8-s + 0.420·10-s − 6.98i·11-s − 3.60·13-s + 42.8i·14-s + 23.9·16-s − 32.3i·17-s + 14.8·19-s − 1.02i·20-s − 24.8·22-s + 14.8i·23-s + ⋯ |
| L(s) = 1 | − 1.77i·2-s − 2.15·4-s + 0.0236i·5-s − 1.72·7-s + 2.05i·8-s + 0.0420·10-s − 0.635i·11-s − 0.277·13-s + 3.06i·14-s + 1.49·16-s − 1.90i·17-s + 0.783·19-s − 0.0510i·20-s − 1.12·22-s + 0.643i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.285919 + 0.552353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.285919 + 0.552353i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 3.60T \) |
| good | 2 | \( 1 + 3.55iT - 4T^{2} \) |
| 5 | \( 1 - 0.118iT - 25T^{2} \) |
| 7 | \( 1 + 12.0T + 49T^{2} \) |
| 11 | \( 1 + 6.98iT - 121T^{2} \) |
| 17 | \( 1 + 32.3iT - 289T^{2} \) |
| 19 | \( 1 - 14.8T + 361T^{2} \) |
| 23 | \( 1 - 14.8iT - 529T^{2} \) |
| 29 | \( 1 + 31.8iT - 841T^{2} \) |
| 31 | \( 1 + 32.3T + 961T^{2} \) |
| 37 | \( 1 + 58.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 0.929T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.46iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 76.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 88.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 0.904T + 5.32e3T^{2} \) |
| 79 | \( 1 - 79.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 15.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8.87T + 9.40e3T^{2} \) |
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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
−12.40403547890870383202164834951, −11.67807653932512180595196105574, −10.55274358552153066532517056427, −9.582391733964147276030594109862, −9.063352700059317765870102508326, −7.08170055592036159125726888540, −5.33494040324363965002573767737, −3.59920750725855100324972515452, −2.75248707098148818890663107347, −0.43821494700938526456860391301,
3.66522591862257357781549448033, 5.25294465841192828351210604098, 6.45616030234700141505289074735, 7.05704913472442954602503392125, 8.429658936897352103329019690691, 9.404390737901276280531163099312, 10.36478720592819452172189024129, 12.62450202606767267510202447227, 12.96667016906561432017883214062, 14.31637859138661761578736306337
