| L(s) = 1 | + i·3-s + 1.82i·5-s − 1.82·7-s + 2·9-s − 0.828i·11-s − i·13-s − 1.82·15-s − 4.65·17-s + 4i·19-s − 1.82i·21-s − 8.82·23-s + 1.65·25-s + 5i·27-s + 8.82i·29-s + 0.828·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.817i·5-s − 0.691·7-s + 0.666·9-s − 0.249i·11-s − 0.277i·13-s − 0.472·15-s − 1.12·17-s + 0.917i·19-s − 0.398i·21-s − 1.84·23-s + 0.331·25-s + 0.962i·27-s + 1.63i·29-s + 0.144·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7378434563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7378434563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 5 | \( 1 - 1.82iT - 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 8.82iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.82iT - 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 + 8.82iT - 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 5.17iT - 61T^{2} \) |
| 67 | \( 1 - 7.65iT - 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 3.65T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6.34T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−10.02354907163118890110362815080, −9.076348243526764758132187281250, −8.238254299528097455698724490886, −7.21644824158907216233763204301, −6.59127902855545637201870813659, −5.81503936452332109942018112851, −4.68493163792132456889761683162, −3.77334512624766813854778824693, −3.09348168342294399763347059067, −1.80615190515842016228699571994,
0.26833654649186200498940252607, 1.65277665069145066084818680929, 2.64190451151985571756019685132, 4.19153831525030593067314375835, 4.55969790256359212312931938483, 5.93779491124039194414442161228, 6.53068074167611397411175894473, 7.37057287951276151483647469564, 8.132211364066242577422623205167, 9.030687500367368281330406125561
