L(s) = 1 | + 0.783i·3-s − 3.56i·5-s − 7-s + 2.38·9-s − 2.51i·11-s + 6.52i·13-s + 2.79·15-s − 1.93·17-s + 6.56i·19-s − 0.783i·21-s − 2.76·23-s − 7.73·25-s + 4.21i·27-s + 2.34i·29-s − 7.50·31-s + ⋯ |
L(s) = 1 | + 0.452i·3-s − 1.59i·5-s − 0.377·7-s + 0.795·9-s − 0.759i·11-s + 1.80i·13-s + 0.721·15-s − 0.470·17-s + 1.50i·19-s − 0.170i·21-s − 0.577·23-s − 1.54·25-s + 0.811i·27-s + 0.435i·29-s − 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.121823931\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.121823931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 0.783iT - 3T^{2} \) |
| 5 | \( 1 + 3.56iT - 5T^{2} \) |
| 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 6.52iT - 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 6.56iT - 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 2.34iT - 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 + 6.05iT - 37T^{2} \) |
| 41 | \( 1 - 7.71T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 - 2.04iT - 53T^{2} \) |
| 59 | \( 1 + 0.911iT - 59T^{2} \) |
| 61 | \( 1 - 2.08iT - 61T^{2} \) |
| 67 | \( 1 - 1.87iT - 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 6.44iT - 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.970830577039380463215741465338, −8.144956072404350811227620009705, −7.35774125596953150404569252698, −6.36248848026720176901888275600, −5.69842199777363958916852935136, −4.77988795402061951843061674093, −4.15747684471866135458237273063, −3.64806567328041044284302776238, −1.96882721105654726705555792375, −1.23933377094262140728374767043,
0.33749829883746251708669183727, 1.97147341423711249101301165023, 2.74175001486549597151397201542, 3.47112208911740204038466139520, 4.46529920082484954724478610728, 5.51834044790632663328342409879, 6.37193010324540970405992047556, 7.00382144682158454196357935762, 7.39583159174722671300740685931, 8.090838334529440610044393604234