L(s) = 1 | − i·3-s + (2.21 + 0.311i)5-s − i·7-s − 9-s + 2.21·11-s + 5.11i·13-s + (0.311 − 2.21i)15-s + 4.11i·17-s + 5.11·19-s − 21-s − i·23-s + (4.80 + 1.37i)25-s + i·27-s + 2.93·29-s − 2.09·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.990 + 0.139i)5-s − 0.377i·7-s − 0.333·9-s + 0.667·11-s + 1.41i·13-s + (0.0803 − 0.571i)15-s + 0.998i·17-s + 1.17·19-s − 0.218·21-s − 0.208i·23-s + (0.961 + 0.275i)25-s + 0.192i·27-s + 0.544·29-s − 0.376·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.131080192\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.131080192\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.21 - 0.311i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 - 5.11iT - 13T^{2} \) |
| 17 | \( 1 - 4.11iT - 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 1.28iT - 37T^{2} \) |
| 41 | \( 1 - 0.458T + 41T^{2} \) |
| 43 | \( 1 + 2.14iT - 43T^{2} \) |
| 47 | \( 1 + 2.40iT - 47T^{2} \) |
| 53 | \( 1 + 3.83iT - 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 1.76iT - 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.4iT - 83T^{2} \) |
| 89 | \( 1 + 0.755T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−9.410305120032169928066386074944, −8.951888477374701974419725162447, −7.87754652019138267660564099973, −6.89256724645819606006035633746, −6.45443942494022937863267200413, −5.57604560422526004081877831011, −4.48107189326942524583628264378, −3.39665812143977080970525631719, −2.08955760222196656314169816222, −1.27926848381581912832397892380,
1.05815017369385823794533804953, 2.58204324289950551159141520302, 3.35442041660878384463288588992, 4.71867043498078344506335484083, 5.45018790962262119970643032311, 6.03882133674771842823554500640, 7.15175100280017397101249278660, 8.086888093081539781884537376003, 9.138347543789432347719735333052, 9.467665499263068100921129172363