| L(s) = 1 | − 1.73i·3-s − 3.46i·7-s − 2.99·9-s + 2·13-s − 3.46i·19-s − 5.99·21-s + 5·25-s + 5.19i·27-s + 10.3i·31-s + 10·37-s − 3.46i·39-s + 10.3i·43-s − 4.99·49-s − 5.99·57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s − 1.30i·7-s − 0.999·9-s + 0.554·13-s − 0.794i·19-s − 1.30·21-s + 25-s + 0.999i·27-s + 1.86i·31-s + 1.64·37-s − 0.554i·39-s + 1.58i·43-s − 0.714·49-s − 0.794·57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799071 - 0.799071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799071 - 0.799071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−12.48156897690249202858230097469, −11.25165812233945499027023204816, −10.60278849114699696559566293376, −9.161734831488399239895785093245, −8.041862585418433570099595947456, −7.11832588898137780415786460288, −6.30023882575175586006277492091, −4.70291808743084588513594320992, −3.10170155764302832632860294951, −1.13039910110545379294649008287,
2.60182485234272276047494052157, 4.02421232543024795597776629105, 5.38158919701726417869764232295, 6.18228802374312921582724731063, 8.031716069625640784140838134814, 8.937434081000418110369817580115, 9.708184890174179423079431268622, 10.84893670739889342900815061984, 11.68598738069570225342057783170, 12.62319681634606304500547076579
