L(s) = 1 | + (1.5 + 2.59i)5-s + (2 − 1.73i)7-s + (1.5 − 2.59i)11-s + 2·13-s + (1.5 − 2.59i)17-s + (−0.5 − 0.866i)19-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + 6·29-s + (−3.5 + 6.06i)31-s + (7.5 + 2.59i)35-s + (0.5 + 0.866i)37-s − 6·41-s + 4·43-s + (4.5 + 7.79i)47-s + ⋯ |
L(s) = 1 | + (0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s + (0.452 − 0.783i)11-s + 0.554·13-s + (0.363 − 0.630i)17-s + (−0.114 − 0.198i)19-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + 1.11·29-s + (−0.628 + 1.08i)31-s + (1.26 + 0.439i)35-s + (0.0821 + 0.142i)37-s − 0.937·41-s + 0.609·43-s + (0.656 + 1.13i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.073309906\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.073309906\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.30358453930121548833682431952, −9.143318187149218585085365487668, −8.333771572038769782727462515938, −7.33646357871022252607207437753, −6.60583695879706074251505012480, −5.85722811949402354590846783750, −4.72339577195609326964576947638, −3.57186241373175757028247360743, −2.62267145068133783731254106507, −1.22306959456710338102849832242,
1.33407431219620289978901744112, 2.14000104244098955872513850263, 3.85856408036476189238154629586, 4.82270177095670116337301578849, 5.56039631665226305706235662355, 6.33112624694184269695350637729, 7.64184450575807198362841769226, 8.439002155969622552887550783153, 9.076406890196947672595574856089, 9.772022525046033312729054674447