| L(s) = 1 | + (−0.801 + 1.83i)2-s + 4.03·3-s + (−2.71 − 2.93i)4-s + (−3.23 + 7.39i)6-s − 11.1i·7-s + (7.55 − 2.62i)8-s + 7.27·9-s + 17.3·11-s + (−10.9 − 11.8i)12-s + 6.70i·13-s + (20.3 + 8.91i)14-s + (−1.25 + 15.9i)16-s + 3.45·17-s + (−5.82 + 13.3i)18-s + 0.787·19-s + ⋯ |
| L(s) = 1 | + (−0.400 + 0.916i)2-s + 1.34·3-s + (−0.678 − 0.734i)4-s + (−0.538 + 1.23i)6-s − 1.58i·7-s + (0.944 − 0.327i)8-s + 0.808·9-s + 1.57·11-s + (−0.912 − 0.987i)12-s + 0.515i·13-s + (1.45 + 0.636i)14-s + (−0.0781 + 0.996i)16-s + 0.203·17-s + (−0.323 + 0.740i)18-s + 0.0414·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85230 + 0.312273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85230 + 0.312273i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.801 - 1.83i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.03T + 9T^{2} \) |
| 7 | \( 1 + 11.1iT - 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 - 6.70iT - 169T^{2} \) |
| 17 | \( 1 - 3.45T + 289T^{2} \) |
| 19 | \( 1 - 0.787T + 361T^{2} \) |
| 23 | \( 1 + 38.1iT - 529T^{2} \) |
| 29 | \( 1 - 37.7iT - 841T^{2} \) |
| 31 | \( 1 - 42.7iT - 961T^{2} \) |
| 37 | \( 1 + 0.378iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.91T + 1.68e3T^{2} \) |
| 43 | \( 1 - 9.21T + 1.84e3T^{2} \) |
| 47 | \( 1 - 31.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 23.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 47.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 59.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 96.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 58.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 44.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 126.T + 9.40e3T^{2} \) |
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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
−12.65307737904569129857966525216, −10.94214343973620754511404375303, −9.948682488115186823603160319270, −9.038161896751899776348269648640, −8.334016774009596879124702208211, −7.17355163679601560162431236444, −6.60450355823580328929858023708, −4.53996354368347792457981881863, −3.62709833095361388118078090740, −1.31617707466093291732192454468,
1.82238673891646934517739248485, 2.92842634896173279157901961491, 3.97893181875894205045956567794, 5.77587301212806498460433071513, 7.62609316209142068570776448696, 8.528983703279533980262853154324, 9.266752600338902969596599031109, 9.736795287459774188352681916797, 11.50065653044802710091673620120, 11.96505269349089372469013671176
