| L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−6.24 − 3.16i)7-s + 2.82·8-s − 2.99·9-s − 1.75·11-s − 3.46i·12-s + 18.7i·13-s + (−8.82 − 4.47i)14-s + 4.00·16-s + 23.4i·17-s − 4.24·18-s + 23.0i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.891 − 0.452i)7-s + 0.353·8-s − 0.333·9-s − 0.159·11-s − 0.288i·12-s + 1.44i·13-s + (−0.630 − 0.319i)14-s + 0.250·16-s + 1.38i·17-s − 0.235·18-s + 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.986073476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986073476\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.24 + 3.16i)T \) |
| good | 11 | \( 1 + 1.75T + 121T^{2} \) |
| 13 | \( 1 - 18.7iT - 169T^{2} \) |
| 17 | \( 1 - 23.4iT - 289T^{2} \) |
| 19 | \( 1 - 23.0iT - 361T^{2} \) |
| 23 | \( 1 + 18.7T + 529T^{2} \) |
| 29 | \( 1 - 30T + 841T^{2} \) |
| 31 | \( 1 + 8.60iT - 961T^{2} \) |
| 37 | \( 1 - 70.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 42.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 51.7iT - 9.40e3T^{2} \) |
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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
−10.04577638993542534057816009869, −9.009140922821490297600769439401, −7.995627063668017421027072460192, −7.21684856811269850843307505636, −6.20023005267359221061046968218, −6.00765617406271250456173378628, −4.36707788518781302707866997637, −3.78801549806953280553052930671, −2.51959211087655151597095553209, −1.39676864579915157196245261553,
0.47762188420151697561238640044, 2.72017514123958501353304404050, 3.04312276767932377745221787067, 4.37813821283638617929305802824, 5.20994963618826235376782018888, 5.96966468695974078392156136641, 6.86709731187403692778341466075, 7.86425304651026707687416265569, 8.839085822500764858689854647658, 9.786773883125724490132508218548
