L(s) = 1 | + (0.273 + 1.38i)2-s + (−1.55 − 0.758i)3-s + (−1.85 + 0.758i)4-s + (0.626 − 2.36i)6-s − 3.56i·7-s + (−1.55 − 2.36i)8-s + (1.85 + 2.36i)9-s + 4.20·11-s + (3.45 + 0.222i)12-s + 2.70·13-s + (4.94 − 0.973i)14-s + (2.85 − 2.80i)16-s − 0.828i·17-s + (−2.77 + 3.21i)18-s − 5.07i·19-s + ⋯ |
L(s) = 1 | + (0.193 + 0.981i)2-s + (−0.899 − 0.437i)3-s + (−0.925 + 0.379i)4-s + (0.255 − 0.966i)6-s − 1.34i·7-s + (−0.550 − 0.834i)8-s + (0.616 + 0.787i)9-s + 1.26·11-s + (0.997 + 0.0642i)12-s + 0.749·13-s + (1.32 − 0.260i)14-s + (0.712 − 0.701i)16-s − 0.200i·17-s + (−0.653 + 0.757i)18-s − 1.16i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.991339 - 0.0318579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.991339 - 0.0318579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.273 - 1.38i)T \) |
| 3 | \( 1 + (1.55 + 0.758i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 5.07iT - 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 5.55iT - 29T^{2} \) |
| 31 | \( 1 - 6.59iT - 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + 0.531iT - 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.701T + 61T^{2} \) |
| 67 | \( 1 - 2.04iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.12iT - 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 - 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−11.78598088701346967877957424199, −10.89864424648501642005511857166, −9.830067157257198648443103854657, −8.683744741153598643136829673568, −7.44945751213071356992752393114, −6.82653652487426727345357804153, −6.04474814931567982762350873513, −4.70914271532875139571447180037, −3.85252059322530232028444401677, −0.900346535699855662153296117377,
1.56862960486840238227799473619, 3.40876520709646186347334872651, 4.47181657406146890817701410400, 5.72893380692458335137116544502, 6.28184847063273008085826292240, 8.343767753871889142743952036898, 9.276319735951225835416826017218, 9.931137944822739982260508234982, 11.13106645817281781871223051581, 11.68029079216577329722913123703