L(s) = 1 | + 0.602i·3-s + i·5-s + 7-s + 2.63·9-s + 5.63i·11-s + 4.43i·13-s − 0.602·15-s − 7.46·17-s + 1.20i·19-s + 0.602i·21-s + 7.27·23-s − 25-s + 3.39i·27-s − 6.67i·29-s − 10.0·31-s + ⋯ |
L(s) = 1 | + 0.347i·3-s + 0.447i·5-s + 0.377·7-s + 0.878·9-s + 1.69i·11-s + 1.22i·13-s − 0.155·15-s − 1.81·17-s + 0.276i·19-s + 0.131i·21-s + 1.51·23-s − 0.200·25-s + 0.653i·27-s − 1.23i·29-s − 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.527415636\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.527415636\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 0.602iT - 3T^{2} \) |
| 11 | \( 1 - 5.63iT - 11T^{2} \) |
| 13 | \( 1 - 4.43iT - 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 1.20iT - 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 + 6.67iT - 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.86iT - 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 2.06iT - 53T^{2} \) |
| 59 | \( 1 + 2.06iT - 59T^{2} \) |
| 61 | \( 1 - 1.27iT - 61T^{2} \) |
| 67 | \( 1 - 7.20iT - 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 1.20iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−9.315900923596962938028257517662, −8.832715794196778102588487784724, −7.41430170203577398746828093839, −7.14718017654993844401612496596, −6.44176141058397403746096248619, −5.11475750009220949515434904714, −4.39358595914105853107362207497, −3.94120775344693292309701218904, −2.33026960688443924944036738347, −1.72843529028177038247500434392,
0.52180139971445660673906628153, 1.57043453676152056929602360160, 2.87186210680163985930002467226, 3.77979129091918002243637389255, 4.89716686071465683846053046585, 5.46328087897822048249583321511, 6.50818668066535631757512903291, 7.17898800640320482668135366640, 8.045251296252818005885259191576, 8.818163993879513980842437763473