L(s) = 1 | − 0.936i·7-s − 2.20i·11-s − 3.33i·13-s − 1.54i·17-s − 3.12·19-s + 3.39·23-s − 8.44·29-s − 8.30i·31-s + 7.60i·37-s + 5.83i·41-s − 7.77·43-s − 10.7·47-s + 6.12·49-s + 5.08·53-s − 10.6i·59-s + ⋯ |
L(s) = 1 | − 0.353i·7-s − 0.665i·11-s − 0.924i·13-s − 0.374i·17-s − 0.716·19-s + 0.707·23-s − 1.56·29-s − 1.49i·31-s + 1.25i·37-s + 0.910i·41-s − 1.18·43-s − 1.56·47-s + 0.874·49-s + 0.698·53-s − 1.39i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2627067685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2627067685\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.936iT - 7T^{2} \) |
| 11 | \( 1 + 2.20iT - 11T^{2} \) |
| 13 | \( 1 + 3.33iT - 13T^{2} \) |
| 17 | \( 1 + 1.54iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 + 8.30iT - 31T^{2} \) |
| 37 | \( 1 - 7.60iT - 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 5.08T + 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 5.59T + 73T^{2} \) |
| 79 | \( 1 - 1.02iT - 79T^{2} \) |
| 83 | \( 1 - 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67027513818570727869926002010, −6.78370305704581513237236746994, −6.17905010646706087531109609195, −5.39491676707641723388450117776, −4.76307223102864856173268261455, −3.78756338902863846395936101189, −3.18461172196972123466378788293, −2.26366926280819510685323242067, −1.11286717266517974835630586689, −0.06436652440949153058797703572,
1.58467050607369350897397069048, 2.14404438397735096041946521377, 3.24171587444243430111768452338, 4.04423350958142489267720804681, 4.75270397413173584963935004077, 5.50369512325729117782879023610, 6.23688560257968112679237003999, 7.08371336778247011214396069018, 7.36597369729889712264484649526, 8.515502012195525237380832607527