L(s) = 1 | + (1.65 − 2.85i)5-s − 1.44·7-s − 1.81·11-s + (0.5 + 0.866i)13-s + (−3.30 + 5.71i)17-s + (−1 + 4.24i)19-s + (−2.39 − 4.14i)23-s + (−2.94 − 5.10i)25-s + (−4.78 − 8.28i)29-s + 4.55·31-s + (−2.39 + 4.14i)35-s − 5.89·37-s + (−1.48 + 2.57i)41-s + (−4.17 + 7.22i)43-s + (−1.48 − 2.57i)47-s + ⋯ |
L(s) = 1 | + (0.738 − 1.27i)5-s − 0.547·7-s − 0.547·11-s + (0.138 + 0.240i)13-s + (−0.800 + 1.38i)17-s + (−0.229 + 0.973i)19-s + (−0.498 − 0.864i)23-s + (−0.589 − 1.02i)25-s + (−0.888 − 1.53i)29-s + 0.817·31-s + (−0.404 + 0.700i)35-s − 0.969·37-s + (−0.231 + 0.401i)41-s + (−0.636 + 1.10i)43-s + (−0.216 − 0.374i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02348562659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02348562659\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1 - 4.24i)T \) |
good | 5 | \( 1 + (-1.65 + 2.85i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.30 - 5.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.39 + 4.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.78 + 8.28i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 + (1.48 - 2.57i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.17 - 7.22i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.48 + 2.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.65 + 2.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.21 + 7.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.17 + 12.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.78 - 8.28i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.17 - 12.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.63T + 83T^{2} \) |
| 89 | \( 1 + (-8.25 - 14.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.44 - 11.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.266965734866769415487674875947, −8.050227086568349412436304638612, −6.48682543230244520565100622742, −6.18199287272461906274534793576, −5.27823369547471872541255678295, −4.46560065056160961854234005121, −3.68554133986650274466078134014, −2.28905909572923242339982342364, −1.52343312024420248087002005180, −0.00685250573126656047620545199,
1.87752698842807773057003658851, 2.85495220483635396581231722797, 3.29821274997912090051984989706, 4.67431763968775905347587566835, 5.53008174700464709458680562306, 6.27831287444203416354210914964, 7.16201299332684730016042545012, 7.29084077826750063332861716369, 8.743125154080273212792002972860, 9.241940203344346019735389213530