| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 0.768i·5-s + 2.64·7-s + 2.82i·8-s + 1.08·10-s + 3.64i·11-s + 4.36·13-s − 3.74i·14-s + 4.00·16-s − 8.55i·17-s − 18.7·19-s − 1.53i·20-s + 5.15·22-s + 12.5i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.153i·5-s + 0.377·7-s + 0.353i·8-s + 0.108·10-s + 0.331i·11-s + 0.335·13-s − 0.267i·14-s + 0.250·16-s − 0.503i·17-s − 0.989·19-s − 0.0768i·20-s + 0.234·22-s + 0.544i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.729884174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729884174\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 0.768iT - 25T^{2} \) |
| 11 | \( 1 - 3.64iT - 121T^{2} \) |
| 13 | \( 1 - 4.36T + 169T^{2} \) |
| 17 | \( 1 + 8.55iT - 289T^{2} \) |
| 19 | \( 1 + 18.7T + 361T^{2} \) |
| 23 | \( 1 - 12.5iT - 529T^{2} \) |
| 29 | \( 1 - 16.4iT - 841T^{2} \) |
| 31 | \( 1 - 13.7T + 961T^{2} \) |
| 37 | \( 1 + 28.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 64.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 49.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 49.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 81.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 8.94iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 90.5T + 9.40e3T^{2} \) |
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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
−9.692431081255693748896795681755, −8.908605514459315733024895246073, −8.154545271715985692920657850981, −7.18792764121882126679565810268, −6.24056033121881442321762217612, −5.11030311659226152690658128098, −4.35813867306051323250305386594, −3.26538044781701593755358726707, −2.23769236757462503394290884815, −1.04620901014228248868755569028,
0.63846001044716556410963551481, 2.16332563866245975644217555850, 3.63612289731147971943209391720, 4.52289522232138568593822884585, 5.44719244038371272918806853492, 6.31357998848239019917015991620, 7.04648542907243783280602744064, 8.150124393117452171219056035190, 8.570155248215958845711175679477, 9.389652034525567956767279346649
