L(s) = 1 | + i·3-s + 1.03i·5-s − 2.44·7-s − 9-s − 5.46i·11-s + 4.24i·13-s − 1.03·15-s + 3.46·17-s + 0.535i·19-s − 2.44i·21-s + 2.82·23-s + 3.92·25-s − i·27-s − 5.93i·29-s + 7.34·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.462i·5-s − 0.925·7-s − 0.333·9-s − 1.64i·11-s + 1.17i·13-s − 0.267·15-s + 0.840·17-s + 0.122i·19-s − 0.534i·21-s + 0.589·23-s + 0.785·25-s − 0.192i·27-s − 1.10i·29-s + 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.428221142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428221142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 5.46iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.93iT - 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 9.14iT - 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.52iT - 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 9.14iT - 61T^{2} \) |
| 67 | \( 1 - 1.07iT - 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + 1.46iT - 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.803571437720965789020813737094, −8.431562142774971706993813842230, −7.29741088926336639745805995878, −6.43522768908301195192860087774, −6.05329317870311327414654255463, −5.04818552433736952923487091917, −4.06413496167661405160533688651, −3.23565513698055294549827711682, −2.73914317915586154752965924193, −1.00880910509107658070971662147,
0.54300852288144398598177176643, 1.71374744279873632203365342915, 2.85184368470712548972350310692, 3.58227953611492674348676261976, 4.91924886202019582295070239189, 5.28909938353552310992088695231, 6.49885837286058980544660591428, 6.93324716790427244129882719454, 7.75737335080699802712059168701, 8.410935985305233516791211335300