| L(s) = 1 | + i·3-s − 1.41i·5-s + 4.24·7-s − 9-s + 6i·11-s + 5.65i·13-s + 1.41·15-s − 6·17-s − 4i·19-s + 4.24i·21-s − 2.82·23-s + 2.99·25-s − i·27-s + 1.41i·29-s − 1.41·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.632i·5-s + 1.60·7-s − 0.333·9-s + 1.80i·11-s + 1.56i·13-s + 0.365·15-s − 1.45·17-s − 0.917i·19-s + 0.925i·21-s − 0.589·23-s + 0.599·25-s − 0.192i·27-s + 0.262i·29-s − 0.254·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.739350662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.739350662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.354645445473843158903176205269, −9.055999623120787120241688236311, −8.199547494583343494725290669488, −7.25707712284242350679201042158, −6.53831527705011687095527797070, −5.05328345852048839220373226192, −4.56658796897671523681622390349, −4.28781478118713908139072943339, −2.31393935281418529212285917885, −1.60181474126501741136921801089,
0.68965853604184405565575827159, 2.03547712999414382622382547343, 3.03358164120905729934056503887, 4.09541131555695810006852240627, 5.41139737298482303099850265332, 5.85231550382732688502712607515, 6.89075077343429651326380444043, 7.895167854410691283736157855780, 8.236053303963820257269288766720, 8.927742756417351200857507592519
