L(s) = 1 | + (0.632 + 1.09i)2-s + (1.31 − 2.27i)3-s + (0.199 − 0.344i)4-s + (−1.45 − 2.51i)5-s + 3.32·6-s + (1.29 − 2.30i)7-s + 3.03·8-s + (−1.95 − 3.37i)9-s + (1.83 − 3.18i)10-s + (1.01 − 1.76i)11-s + (−0.523 − 0.906i)12-s + (3.34 − 0.0400i)14-s − 7.62·15-s + (1.52 + 2.63i)16-s + (−1.99 + 3.46i)17-s + (2.46 − 4.27i)18-s + ⋯ |
L(s) = 1 | + (0.447 + 0.775i)2-s + (0.758 − 1.31i)3-s + (0.0995 − 0.172i)4-s + (−0.649 − 1.12i)5-s + 1.35·6-s + (0.489 − 0.871i)7-s + 1.07·8-s + (−0.650 − 1.12i)9-s + (0.580 − 1.00i)10-s + (0.307 − 0.531i)11-s + (−0.151 − 0.261i)12-s + (0.894 − 0.0107i)14-s − 1.96·15-s + (0.380 + 0.659i)16-s + (−0.484 + 0.839i)17-s + (0.582 − 1.00i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0513 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.879608542\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.879608542\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.29 + 2.30i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.632 - 1.09i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.31 + 2.27i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.45 + 2.51i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 1.76i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.99 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.48 - 6.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.313 - 0.543i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + (5.21 - 9.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.54 + 2.67i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.521T + 41T^{2} \) |
| 43 | \( 1 - 0.329T + 43T^{2} \) |
| 47 | \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.01 + 1.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.20 + 2.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.34 + 12.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 + (-1.48 + 2.57i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.38 - 7.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.32T + 97T^{2} \) |
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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
−9.130761280563739446430484355370, −8.272078103546286233966402443894, −7.85699072927449188737024407749, −7.20596332557231651325284303206, −6.37502112387341545178616589416, −5.42051698488800254333919754071, −4.41950386254841148427017012609, −3.53017543353884194571818430951, −1.66794665270985384190261762004, −1.10190866919114607791222693522,
2.33080952523630176432825047521, 2.84468637507219491462951446611, 3.70243396208322382864115656749, 4.46237112823602587210143850068, 5.24601987594226285615840186975, 6.90590864697590532003273779603, 7.52233242898347853603450285669, 8.544453781937748106285746655588, 9.329004783249085351373695015247, 10.04905902152423586562456881086