L(s) = 1 | + (−0.696 + 1.87i)2-s + (−3.02 − 2.61i)4-s − 5.46·7-s + (7.00 − 3.86i)8-s + 11.0i·11-s + 10.1i·13-s + (3.80 − 10.2i)14-s + (2.35 + 15.8i)16-s − 24.4i·17-s + 23.7i·19-s + (−20.6 − 7.69i)22-s + 37.2·23-s + (−18.9 − 7.05i)26-s + (16.5 + 14.2i)28-s − 25.7·29-s + ⋯ |
L(s) = 1 | + (−0.348 + 0.937i)2-s + (−0.757 − 0.652i)4-s − 0.781·7-s + (0.875 − 0.482i)8-s + 1.00i·11-s + 0.778i·13-s + (0.272 − 0.732i)14-s + (0.147 + 0.989i)16-s − 1.43i·17-s + 1.25i·19-s + (−0.940 − 0.349i)22-s + 1.61·23-s + (−0.730 − 0.271i)26-s + (0.591 + 0.510i)28-s − 0.888·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01693088338\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01693088338\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.696 - 1.87i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 5.46T + 49T^{2} \) |
| 11 | \( 1 - 11.0iT - 121T^{2} \) |
| 13 | \( 1 - 10.1iT - 169T^{2} \) |
| 17 | \( 1 + 24.4iT - 289T^{2} \) |
| 19 | \( 1 - 23.7iT - 361T^{2} \) |
| 23 | \( 1 - 37.2T + 529T^{2} \) |
| 29 | \( 1 + 25.7T + 841T^{2} \) |
| 31 | \( 1 - 4.83iT - 961T^{2} \) |
| 37 | \( 1 + 35.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 55.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 104.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 49.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 72.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 72.9iT - 9.40e3T^{2} \) |
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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
−9.492346765614926121685886582622, −8.994679061996194631919138370004, −7.78054303480989809625162778408, −7.06075350996262127187669438449, −6.48509109742579317821740846274, −5.33799307010199411437253250967, −4.56575500805733373193654681610, −3.34775902706731697678265277531, −1.68683072704870813401642982390, −0.00671605632877191305171607955,
1.27937828403310063930989669324, 2.89777202384451221275746461501, 3.39466370777096524429337265333, 4.65603708694866943933988531468, 5.74422901355740192186520661621, 6.79441913384687766504920088263, 7.906623383602995573092587168191, 8.711968928828442325020699917990, 9.323669173454291561609185941075, 10.30897721844371737535575514332