| L(s) = 1 | − i·2-s − 4-s − 1.68·5-s + i·8-s + 1.68i·10-s − 3.32i·11-s + 4.34i·13-s + 16-s + 3.81·17-s − 4.06i·19-s + 1.68·20-s − 3.32·22-s − 4.69i·23-s − 2.16·25-s + 4.34·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.753·5-s + 0.353i·8-s + 0.532i·10-s − 1.00i·11-s + 1.20i·13-s + 0.250·16-s + 0.924·17-s − 0.932i·19-s + 0.376·20-s − 0.709·22-s − 0.979i·23-s − 0.432·25-s + 0.852·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5780814750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5780814750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.68T + 5T^{2} \) |
| 11 | \( 1 + 3.32iT - 11T^{2} \) |
| 13 | \( 1 - 4.34iT - 13T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 + 4.06iT - 19T^{2} \) |
| 23 | \( 1 + 4.69iT - 23T^{2} \) |
| 29 | \( 1 - 2.50iT - 29T^{2} \) |
| 31 | \( 1 - 4.09iT - 31T^{2} \) |
| 37 | \( 1 - 6.27T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.412T + 47T^{2} \) |
| 53 | \( 1 + 9.09iT - 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 13.9iT - 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 + 4.94iT - 71T^{2} \) |
| 73 | \( 1 - 5.78iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−8.519380285916942877210918647925, −7.921126552917145916162390770021, −6.94928938785914786396197891365, −6.17560513846053340470461577398, −5.09246351849578023516023464905, −4.34433705477903544309519714203, −3.51777273451322324520251009867, −2.75804766785037669557599827885, −1.47125368798657787563951326978, −0.20646726011263744404295227837,
1.33338622094720583021153941667, 2.88240397560595590986842386669, 3.83765939206543539691771913851, 4.52732775372424181045496608345, 5.58782842176647911351424736287, 6.03002926339328834540964293382, 7.29885152952417457959070895072, 7.72068202251550327563563110466, 8.106633598883076303896817429956, 9.218342576918132282157308059805
