L(s) = 1 | + (1 + 1.41i)3-s + (1.73 + 1.41i)5-s + (−1 + 2.44i)7-s + (−1.00 + 2.82i)9-s − 2.82i·11-s − 4·13-s + (−0.267 + 3.86i)15-s + 2.82i·17-s + (−4.46 + 1.03i)21-s − 3.46·23-s + (0.999 + 4.89i)25-s + (−5.00 + 1.41i)27-s − 5.65i·29-s + 9.79i·31-s + (4.00 − 2.82i)33-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s + (0.774 + 0.632i)5-s + (−0.377 + 0.925i)7-s + (−0.333 + 0.942i)9-s − 0.852i·11-s − 1.10·13-s + (−0.0691 + 0.997i)15-s + 0.685i·17-s + (−0.974 + 0.225i)21-s − 0.722·23-s + (0.199 + 0.979i)25-s + (−0.962 + 0.272i)27-s − 1.05i·29-s + 1.75i·31-s + (0.696 − 0.492i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.713518835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.713518835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 4.89iT - 67T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.809057975393415069481374329790, −8.946631362442555688364219941338, −8.373570561457750405704824797375, −7.37582863862328833364285136017, −6.26227359027486265947535164916, −5.67302179039113166544948231310, −4.80933206630686428590201149847, −3.57528965354390026691034031603, −2.79380337952894947239208656429, −2.05456448705993453431642946591,
0.55656417443746608427149966215, 1.84583168434823179408133003449, 2.64901101061555593697206518937, 3.94917735675229672650191967727, 4.85567669618632971915874273471, 5.87453257903040793714271481088, 6.83095403057336439697985156169, 7.37116210056843074860561631816, 8.093892644842734210176082969846, 9.166456923334096161481572225702