| 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
| \(L(1)\) |
\(\approx\) |
\(2.013114253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013114253\) |
| \(L(\frac{3}{2})\) |
|
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
−9.224495297065983412175736417356, −8.766915103619767284264409567024, −7.88411604219452976815487352505, −7.03841279147405011291846260153, −6.02170081358184721245694996384, −4.93706443314795119455595915785, −4.55409126448597055793321081573, −3.64339351356686632142130673678, −2.18016300308326187144496956476, −1.00584802973442929444495773786,
1.10993560892171497698362763882, 2.27622455078625036309342255901, 3.18803340054070142600587721623, 4.51531206893552616575774820732, 5.13067665464011777641277480322, 6.50715947781835614679005544326, 6.99263132645719490536824334116, 7.50292658493509646956408853473, 8.597494349157749621647714498634, 9.219024705022413816986815223073
