L(s) = 1 | + (−0.5 + 0.866i)5-s + (2 − 1.73i)7-s + (−1.5 − 2.59i)11-s − 6·13-s + (−2.5 − 4.33i)17-s + (0.5 − 0.866i)19-s + (3.5 − 6.06i)23-s + (2 + 3.46i)25-s − 2·29-s + (−2.5 − 4.33i)31-s + (0.499 + 2.59i)35-s + (−1.5 + 2.59i)37-s + 2·41-s + 4·43-s + (−2.5 + 4.33i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.452 − 0.783i)11-s − 1.66·13-s + (−0.606 − 1.05i)17-s + (0.114 − 0.198i)19-s + (0.729 − 1.26i)23-s + (0.400 + 0.692i)25-s − 0.371·29-s + (−0.449 − 0.777i)31-s + (0.0845 + 0.439i)35-s + (−0.246 + 0.427i)37-s + 0.312·41-s + 0.609·43-s + (−0.364 + 0.631i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028006477\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028006477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (2.5 - 4.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.710601953150148855921745639134, −8.945264058360348999736912160310, −7.81259332042344261872049267929, −7.36538705080999696288330089808, −6.48883265786195081405065911330, −5.08722790195235997598837495247, −4.65339369398508270856883753113, −3.25205905068220530516479109764, −2.28650678849815266617179821225, −0.45145050942181522872673004287,
1.70523669499749284534433395274, 2.69182966712179297451851881764, 4.22951617286718871558987262583, 4.98919013874357958876284647076, 5.69051225049821569371642582372, 7.08972686251183999837737529079, 7.65560837761174788327332283930, 8.593158643428850140872686802970, 9.293457820324809242173529516567, 10.19798157782018873181031389612