| L(s) = 1 | + 5.14·3-s + 2.23i·5-s − 7.88i·7-s + 17.4·9-s + 3.98i·11-s + 13.5·13-s + 11.4i·15-s + 8.06i·17-s + 2.96i·19-s − 40.5i·21-s + (4.80 − 22.4i)23-s − 5.00·25-s + 43.3·27-s + 0.688·29-s + 36.6·31-s + ⋯ |
| L(s) = 1 | + 1.71·3-s + 0.447i·5-s − 1.12i·7-s + 1.93·9-s + 0.362i·11-s + 1.03·13-s + 0.766i·15-s + 0.474i·17-s + 0.156i·19-s − 1.92i·21-s + (0.208 − 0.977i)23-s − 0.200·25-s + 1.60·27-s + 0.0237·29-s + 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.349457162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.349457162\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-4.80 + 22.4i)T \) |
| good | 3 | \( 1 - 5.14T + 9T^{2} \) |
| 7 | \( 1 + 7.88iT - 49T^{2} \) |
| 11 | \( 1 - 3.98iT - 121T^{2} \) |
| 13 | \( 1 - 13.5T + 169T^{2} \) |
| 17 | \( 1 - 8.06iT - 289T^{2} \) |
| 19 | \( 1 - 2.96iT - 361T^{2} \) |
| 29 | \( 1 - 0.688T + 841T^{2} \) |
| 31 | \( 1 - 36.6T + 961T^{2} \) |
| 37 | \( 1 + 3.63iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 42.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 25.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 31.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 108. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 67.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 48.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 73.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 92.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138. iT - 9.40e3T^{2} \) |
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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
−8.712200363823363393400841627589, −8.470918174381406261888889863064, −7.46294673841193787840888492407, −7.02556858054810045722733650396, −6.05563456338193815760419275818, −4.46835689066900331821209084578, −3.90287483320376572566132677280, −3.13556751099440337997366592783, −2.17745669695260543558092978221, −1.06696604529434903378568023510,
1.21001995368578850735100020907, 2.27015812178143029828743128258, 3.07495152180283698698366791758, 3.83070582913036717380008518803, 4.92269214981458664168508118197, 5.87480948187009693084525604903, 6.85699814610576492762675474507, 7.964545678723513267024129895334, 8.328538104827299708339421930737, 9.111213889090077960801943344221
