| L(s) = 1 | − 2.56i·3-s + 4.56i·7-s − 3.56·9-s + 11-s − 1.12i·13-s + 7.68i·17-s + 1.43·19-s + 11.6·21-s − 1.12i·23-s + 1.43i·27-s − 8.56·29-s + 1.43·31-s − 2.56i·33-s − 7.43i·37-s − 2.87·39-s + ⋯ |
| L(s) = 1 | − 1.47i·3-s + 1.72i·7-s − 1.18·9-s + 0.301·11-s − 0.311i·13-s + 1.86i·17-s + 0.330·19-s + 2.54·21-s − 0.234i·23-s + 0.276i·27-s − 1.58·29-s + 0.258·31-s − 0.445i·33-s − 1.22i·37-s − 0.460·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4480105244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4480105244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 7 | \( 1 - 4.56iT - 7T^{2} \) |
| 13 | \( 1 + 1.12iT - 13T^{2} \) |
| 17 | \( 1 - 7.68iT - 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 1.12iT - 23T^{2} \) |
| 29 | \( 1 + 8.56T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 7.43iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 3.12iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 9.68iT - 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 1.12iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 - 4.87iT - 97T^{2} \) |
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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
−8.555587073749237883490923048172, −7.890416649521050963456829591485, −7.16188583935630000803139902434, −6.29506493531925416759042479262, −5.87717100963176116918991198933, −5.26944268836679304414279811588, −3.92056045806551234893046906379, −2.92600853117168278623526326204, −2.00970616660984152801452660396, −1.53379312219997006394974933559,
0.11961166735232015573179692954, 1.45421478623777377394555206888, 3.10484974676912299935959354639, 3.56409314679268380447318305848, 4.51110737315994694235578976882, 4.75995703646879142740097764448, 5.71317582909025468947162493994, 6.91214363628668362467597124978, 7.24807744912913377385068789324, 8.167124646737807630690041122072
