L(s) = 1 | + i·2-s − 4-s − 2i·7-s − i·8-s + 3·11-s + 2i·13-s + 2·14-s + 16-s − 3i·17-s + 19-s + 3i·22-s + 6i·23-s − 2·26-s + 2i·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s + 0.904·11-s + 0.554i·13-s + 0.534·14-s + 0.250·16-s − 0.727i·17-s + 0.229·19-s + 0.639i·22-s + 1.25i·23-s − 0.392·26-s + 0.377i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.844695765\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844695765\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−8.443682876602211785625774750278, −7.64020727177535560718338850635, −6.82942766353346685523230801964, −6.66580189105063232319291758141, −5.47335187325497441637113811051, −4.86230117149344949777863060800, −3.92493503189121857420431104568, −3.35991565849462132810090556189, −1.86068554715974495810167109786, −0.74374774707116788588238715102,
0.857266828283145385884564880879, 1.96765964058205846150258867384, 2.81319050715430312872585402958, 3.68700068019899057794926897598, 4.48241112114823945773560563889, 5.35016854750533457278605774730, 6.11115844245026357242835008025, 6.81243767401715786193774743583, 7.88437008125102159658082766725, 8.598757719684644525004190562698