L(s) = 1 | + (0.119 − 0.207i)5-s + (0.710 + 2.54i)7-s + (1.21 + 2.09i)11-s + 0.760·13-s + (−1.71 − 2.96i)17-s + (−0.590 + 1.02i)19-s + (−1.09 + 1.88i)23-s + (2.47 + 4.28i)25-s + 2.89·29-s + (2.32 + 4.02i)31-s + (0.612 + 0.157i)35-s + (−2.89 + 5.00i)37-s − 8.54·41-s − 3.37·43-s + (2.58 − 4.47i)47-s + ⋯ |
L(s) = 1 | + (0.0534 − 0.0926i)5-s + (0.268 + 0.963i)7-s + (0.364 + 0.632i)11-s + 0.211·13-s + (−0.414 − 0.718i)17-s + (−0.135 + 0.234i)19-s + (−0.227 + 0.394i)23-s + (0.494 + 0.856i)25-s + 0.538·29-s + (0.417 + 0.722i)31-s + (0.103 + 0.0266i)35-s + (−0.475 + 0.823i)37-s − 1.33·41-s − 0.515·43-s + (0.376 − 0.652i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.571964027\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.571964027\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.710 - 2.54i)T \) |
good | 5 | \( 1 + (-0.119 + 0.207i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.760T + 13T^{2} \) |
| 17 | \( 1 + (1.71 + 2.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.590 - 1.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 - 1.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + (-2.32 - 4.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.89 - 5.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + (-2.58 + 4.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.56 + 2.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.11 - 5.39i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.681 - 1.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.03 - 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + (-3.91 - 6.77i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.27 - 3.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + (2.73 - 4.74i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.605945475663119389293475416362, −8.779493444749344888992243172370, −8.273181839343894117800494355353, −7.10915659276531881258923782753, −6.50436366105192128891316498523, −5.36101777214188702659672372067, −4.84275919853703934889760582857, −3.61843871458610222786406055279, −2.52776863876765964664147453265, −1.45519481291397627519341149704,
0.64806325990727449707328948672, 2.00147238675535366587906015662, 3.35336153208492440410585512588, 4.17431520608749698858982480075, 5.02034191653548382110640043740, 6.29942325097310981743748403356, 6.67489082005918416498432148806, 7.83564478263857366314274179981, 8.402743936248046716819850081763, 9.253948519312525161695660958416