| L(s) = 1 | − i·7-s + 3·9-s − 6·11-s − 2i·13-s + 3i·17-s + 6·19-s + i·23-s − 3·29-s − 3·31-s − i·37-s + 9·41-s − 8i·43-s − 4i·47-s + 6·49-s + i·53-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s + 9-s − 1.80·11-s − 0.554i·13-s + 0.727i·17-s + 1.37·19-s + 0.208i·23-s − 0.557·29-s − 0.538·31-s − 0.164i·37-s + 1.40·41-s − 1.21i·43-s − 0.583i·47-s + 0.857·49-s + 0.137i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633242703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633242703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−7.974684030038371362661499655364, −7.46721978945700448306441662723, −7.05327383951325524737312396510, −5.70561057142759781889338646595, −5.42755694932790245043144064116, −4.43718168230733038584775330490, −3.63685219384694221354548819077, −2.75557830537893932966884538548, −1.73684536868741167577433724392, −0.52597335362696331861488673489,
0.992050436900914754116202054484, 2.23419283449759969299113854842, 2.91103777254490309407546115060, 3.96414004902763441188966460948, 4.90217461302165383951306066785, 5.35202882263727020787012978441, 6.23306749822150298562826978274, 7.32455680407539074097898174167, 7.50760568985551690456235254807, 8.332500620763886995163268329317
