L(s) = 1 | + (1.11 − 1.32i)3-s + (0.439 + 0.761i)5-s + (0.5 − 0.866i)7-s + (−0.520 − 2.95i)9-s + (1.93 − 3.35i)11-s + (2.72 + 4.72i)13-s + (1.5 + 0.264i)15-s + 1.65·17-s − 2.41·19-s + (−0.592 − 1.62i)21-s + (1.58 + 2.73i)23-s + (2.11 − 3.66i)25-s + (−4.5 − 2.59i)27-s + (3.02 − 5.23i)29-s + (−2.27 − 3.94i)31-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)3-s + (0.196 + 0.340i)5-s + (0.188 − 0.327i)7-s + (−0.173 − 0.984i)9-s + (0.584 − 1.01i)11-s + (0.756 + 1.30i)13-s + (0.387 + 0.0682i)15-s + 0.400·17-s − 0.553·19-s + (−0.129 − 0.355i)21-s + (0.329 + 0.571i)23-s + (0.422 − 0.732i)25-s + (−0.866 − 0.499i)27-s + (0.561 − 0.972i)29-s + (−0.408 − 0.708i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.201575050\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.201575050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.11 + 1.32i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.439 - 0.761i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 3.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.02 + 5.23i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.27 + 3.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 + (-0.592 - 1.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0923 + 0.160i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.511 - 0.885i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 + (-3.33 - 5.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.29 + 2.24i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.47 + 2.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + (2.97 - 5.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.109 - 0.189i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + (6.25 - 10.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.647581851045754290252241054099, −8.824546491935362081186182693333, −8.290703276922955241009827101882, −7.24099846412641456310032281346, −6.50419620413243778186133385311, −5.88166780282746217921209615962, −4.24306421836378867503753801782, −3.45148047845483548542918218645, −2.24280527903032127388952740519, −1.07557105603813125082445035523,
1.54739200221319459854699463644, 2.87379612677785301862097154924, 3.81649840934088093443857162008, 4.89601658221015665136827576233, 5.51106899598835049743640217072, 6.79934270521761139760556170017, 7.79635229867851623251652596321, 8.750660061213648832307443604644, 9.029739825518235027177698170882, 10.25143137163148994206572070459