L(s) = 1 | + (0.856 + 1.12i)2-s + (−0.533 + 1.92i)4-s − 3.85i·5-s + (−2.62 + 1.05i)8-s + (4.33 − 3.30i)10-s + 1.36i·11-s − 0.369i·13-s + (−3.43 − 2.05i)16-s − 4.50i·17-s + 0.0661·19-s + (7.43 + 2.05i)20-s + (−1.53 + 1.16i)22-s − 3.20i·23-s − 9.86·25-s + (0.416 − 0.316i)26-s + ⋯ |
L(s) = 1 | + (0.605 + 0.795i)2-s + (−0.266 + 0.963i)4-s − 1.72i·5-s + (−0.928 + 0.371i)8-s + (1.37 − 1.04i)10-s + 0.410i·11-s − 0.102i·13-s + (−0.857 − 0.513i)16-s − 1.09i·17-s + 0.0151·19-s + (1.66 + 0.459i)20-s + (−0.326 + 0.248i)22-s − 0.669i·23-s − 1.97·25-s + (0.0816 − 0.0621i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.555914389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555914389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.856 - 1.12i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.85iT - 5T^{2} \) |
| 11 | \( 1 - 1.36iT - 11T^{2} \) |
| 13 | \( 1 + 0.369iT - 13T^{2} \) |
| 17 | \( 1 + 4.50iT - 17T^{2} \) |
| 19 | \( 1 - 0.0661T + 19T^{2} \) |
| 23 | \( 1 + 3.20iT - 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 + 8.45iT - 41T^{2} \) |
| 43 | \( 1 + 6.30iT - 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 + 1.43iT - 61T^{2} \) |
| 67 | \( 1 + 9.76iT - 67T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 1.80iT - 73T^{2} \) |
| 79 | \( 1 - 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.29iT - 89T^{2} \) |
| 97 | \( 1 + 2.88iT - 97T^{2} \) |
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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
−9.005736988844005638849892790449, −8.358611483616880137654193023192, −7.54981761909109441498538831856, −6.78935306440907685961774203498, −5.65791904737199159580999508525, −5.07635700421815816717577611960, −4.48794587634441519811641789090, −3.53801299447933379920698515792, −2.10541714479514693820332366545, −0.45945530320451368710727463159,
1.61280496001875566236065744641, 2.70403043588422651951002433818, 3.40193995091083498163749424802, 4.14282389364926855301877005279, 5.43720647656412618744520180952, 6.19849620928523269410700356786, 6.79842971913750243797004369651, 7.76200249652753611850736432627, 8.819869657773195552279910779183, 9.819279197821553820449285804460