L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 1.73i·6-s + (2 + 3.46i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−1.5 − 2.59i)11-s + (−1.49 + 0.866i)12-s + (2 − 3.46i)13-s + (−1.99 + 3.46i)14-s + (−0.5 − 0.866i)16-s − 3·17-s + (−1.5 + 2.59i)18-s − 4·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.707i·6-s + (0.755 + 1.30i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.452 − 0.783i)11-s + (−0.433 + 0.250i)12-s + (0.554 − 0.960i)13-s + (−0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s − 0.727·17-s + (−0.353 + 0.612i)18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42726 + 1.70095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42726 + 1.70095i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.14314340690204897746785475618, −10.55519244422138762913497176099, −9.066275175637467104173195434364, −8.516292674807413646601919492850, −8.074169004599430264829075125345, −6.63661319606526328481181839060, −5.46079380721728476064883568850, −4.76714527921270499880497231625, −3.36781666777848659652600868573, −2.34412185611839136063214174642,
1.34955636937724162971654697395, 2.45413470281923141491536517104, 4.05473827716995837511463240104, 4.46505986533899908881375319689, 6.27593653342452062053381268112, 7.30172789688180291354122092982, 8.020053528427071814707162149594, 9.164694220557071644162259824268, 9.937741273579913439812830236781, 11.02870822182463731242825831331